#500307

order	512
nilpotency class	4
isoclinism	isoclinic class of #500307
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#3227
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#24061
prime radical	#60685	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

761 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 761 conjugacy classes of subgroups — too many to draw. Only its 150 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)(3,9)(4,19)(5,25)(6,12)(7,31)(8,13)(10,22)(11,30)(14,43)(15,33)(16,47)(17,35)(18,51)(20,55)(21,53)(23,54)(24,40)(26,59)(27,38)(28,60)(29,41)(32,45)(34,50)(36,48)(37,56)(39,49)(42,58)(44,63)(46,52)(57,62)(61,64);(1,3)(2,9)(4,20,21,46)(5,26)(6,17)(7,18)(8,15)(10,36,37,62)(11,39)(12,35)(13,33)(14,23,44,24)(16,28)(19,52,53,55)(22,57,56,48)(25,59)(27,45,58,64)(29,50)(30,49)(31,51)(32,38,61,42)(34,41)(40,63,54,43)(47,60);(1,4,7,23,8,21,6,24)(2,10,12,42,13,37,31,38)(3,14,18,20,15,44,17,46)(5,27,30,56,28,58,29,22)(9,32,35,62,33,61,51,36)(11,40,25,53,41,54,60,19)(16,48,50,64,26,57,49,45)(34,52,47,63,39,55,59,43);(3,15)(5,28)(9,33)(11,41)(14,44)(17,18)(19,53)(20,46)(22,56)(25,60)(27,58)(29,30)(32,61)(35,51)(36,62)(40,54)

Polycyclic presentation:

2^9|b^2=e,c^2=eh,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=cg,d^b=di,f^b=fi,e^c=ei,f^c=fhi,g^c=gi,h^c=hi,f^d=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307828, #72307907, #72307945
maximal normal subgroup of	#72307828, #72307907, #72307945

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 407503);