#500744

order	512
nilpotency class	4
isoclinism	isoclinic class of #500743
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#24248
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2929 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1022 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

2^9|b^2=e,d^2=ei,f^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=ghi,h^a=hi,c^b=cg,f^b=fh,g^b=gh,h^b=hi,d^c=dh,e^c=ei,f^c=fh,g^c=ghi,h^c=hi,g^d=gi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307115, #72307124, C₂×#500744
maximal normal subgroup of	#72307115, #72307124, C₂×#500744
maximal quotient of	C₂×#500744

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 407940);