#510876

order	512
nilpotency class	4
isoclinism	isoclinic class of #509118
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#3199
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#32004
prime radical	#28180	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #2929 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

438 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=efh,e^2=i,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,e^b=ei,g^b=gh,h^b=hi,f^c=fi,g^c=gh,h^c=hi,e^d=ei,f^d=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 418072);