#7624858

order	512
nilpotency class	4
isoclinism	isoclinic class of #7624824
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁵
prime radical	#532	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #532 ◁ #2931 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1263 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 72 points:

(1,2)(3,7)(4,6)(5,8)(9,10,16,21)(11,17,25,41)(12,26,31,55)(13,19,15,36)(14,34,35,61)(18,42,44,29)(20,33,45,27)(22,46,50,70)(23,38,32,40)(24,51,52,66)(28,53,30,59)(37,62,65,49)(39,60,67,47)(43,56,54,58)(48,68,57,69)(63,72,64,71);(1,3)(2,5)(4,8)(6,7)(9,11)(10,17)(12,22)(13,32)(14,24)(15,23)(16,25)(18,37)(19,40)(20,39)(21,41)(26,46)(27,47)(28,57)(29,49)(30,48)(31,50)(33,60)(34,51)(35,52)(36,38)(42,62)(43,64)(44,65)(45,67)(53,69)(54,63)(55,70)(56,71)(58,72)(59,68)(61,66);(9,12,27,56,15,30,35,42,16,31,33,58,13,28,14,29)(10,18,34,53,19,43,45,55,21,44,61,59,36,54,20,26)(11,22,47,71,23,48,52,62,25,50,60,72,32,57,24,49)(17,37,51,69,40,64,67,70,41,65,66,68,38,63,39,46);(3,8)(5,7)(9,11)(10,17)(12,22)(13,32)(14,24)(15,23)(16,25)(18,37)(19,40)(20,39)(21,41)(26,46)(27,47)(28,57)(29,49)(30,48)(31,50)(33,60)(34,51)(35,52)(36,38)(42,62)(43,64)(44,65)(45,67)(53,69)(54,63)(55,70)(56,71)(58,72)(59,68)(61,66);(2,6)(5,7)(9,13,16,15)(10,19,21,36)(11,23,25,32)(12,28,31,30)(14,33,35,27)(17,38,41,40)(18,43,44,54)(20,34,45,61)(22,48,50,57)(24,47,52,60)(26,53,55,59)(29,58,42,56)(37,63,65,64)(39,66,67,51)(46,68,70,69)(49,71,62,72)

Polycyclic presentation:

2^9|a^2=i,c^2=gh,e^2=i,g^2=hi,h^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gh,h^a=hi,d^b=df,e^b=ei,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7532054);