#7624875

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
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #532 ◁ #2931 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1287 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|b^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,c^b=ci,d^b=df,e^b=ei,d^c=di,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, 7532071);