#484651

order	512
nilpotency class	4
isoclinism	isoclinic class of #12170
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1598
commutator, Frattini, cosocle	C₂×M₁₆	C₂⁴
socle	C₂	C₂×#1598
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 881 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)(3,14)(4,19)(5,11)(6,25)(7,30)(8,13)(9,17)(10,22)(12,29)(15,43)(16,41)(18,42)(20,50)(21,53)(23,36)(24,39)(26,33)(27,57)(28,58)(31,38)(32,47)(34,48)(35,60)(37,52)(40,46)(44,63)(45,49)(51,55)(54,62)(56,64)(59,61);(1,3)(2,9)(4,15)(5,16)(6,26)(7,31)(8,18)(10,32)(11,33)(12,38)(13,34)(14,25)(17,29)(19,49)(20,44)(21,54)(22,46)(23,56)(24,48)(27,59)(28,60)(30,35)(36,61)(37,63)(39,41)(40,57)(42,58)(43,53)(45,55)(47,52)(50,62)(51,64);(1,4,8,22)(2,10)(3,15,18,46)(5,20,7,23)(6,27,28,21)(9,32)(11,36)(12,37)(13,19)(14,40)(16,44,31,56)(17,47,48,45)(24,55,29,52)(25,57)(26,59,60,54)(30,50)(33,61)(34,49)(35,62)(38,63)(39,51)(41,64)(42,43)(53,58);(1,5,8,7)(2,11,13,30)(3,16,18,31)(4,20,22,23)(6,24,28,29)(9,33,34,35)(10,36,19,50)(12,25,39,58)(14,41,42,38)(15,44,46,56)(17,26,48,60)(21,52,27,55)(32,61,49,62)(37,57,51,53)(40,64,43,63)(45,54,47,59)

Polycyclic presentation:

2^9|c^2=h,d^2=i,e^2=gi,g^2=i,b^a=be,c^a=cf,e^a=eg,g^a=gi,h^a=hi,e^b=eg,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	#72281031, #72281057, #72281192, …
maximal normal subgroup of	#72281031, #72281057, #72281192, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 391847);