#484575

order	512
nilpotency class	4
isoclinism	isoclinic class of #12301
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1613
commutator, Frattini, cosocle	C₂×M₁₆	C₂⁴
socle	C₂	C₂×#1613
prime radical	#60749	C₂
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

819 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 819 conjugacy classes of subgroups — too many to draw. Only its 136 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

(1,2,7,13,8,14,5,11)(3,15,17,37,20,44,19,43)(4,21,24,10,25,40,22,36)(6,27,29,12,30,41,26,39)(9,28,32,18,34,49,33,50)(16,45,46,42,48,63,51,60)(23,38,56,53,57,35,54,58)(31,59,52,55,62,47,61,64);(1,3)(2,9)(4,16)(5,17)(6,28)(7,19)(8,20)(10,31)(11,32)(12,37)(13,33)(14,34)(15,41)(18,26)(21,52)(22,46)(23,55)(24,51)(25,48)(27,44)(29,50)(30,49)(35,45)(36,62)(38,63)(39,43)(40,61)(42,53)(47,54)(56,59)(57,64)(58,60);(1,4)(2,10)(3,16)(5,22)(6,23)(7,24)(8,25)(9,31)(11,21)(12,38)(13,40)(14,36)(15,42)(17,46)(18,47)(19,51)(20,48)(26,54)(27,58)(28,55)(29,56)(30,57)(32,52)(33,61)(34,62)(35,39)(37,63)(41,53)(43,45)(44,60)(49,64)(50,59);(1,5,8,7)(2,11,14,13)(3,17,20,19)(4,22,25,24)(6,26,30,29)(9,32,34,33)(10,21,36,40)(12,27,39,41)(15,37,44,43)(16,46,48,51)(18,49,50,28)(23,54,57,56)(31,52,62,61)(35,53,38,58)(42,63,60,45)(47,64,59,55)

Polycyclic presentation:

2^9|a^2=gh,d^2=i,e^2=g,f^2=i,g^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fi,g^a=gi,h^a=hi,e^b=egi,f^b=fhi,g^b=gi,e^c=eh,f^c=fi,f^e=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307084, #72307227, #72307445, …
maximal normal subgroup of	#72307084, #72307227, #72307445, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 391771);