#7623175

order	512
nilpotency class	4
isoclinism	isoclinic class of #7622976
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁵
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1610 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)(3,6)(4,5)(7,8)(9,10,16,22)(11,23,25,50)(12,29,30,60)(13,20,35,48)(14,19,36,47)(15,37,34,21)(17,38,40,26)(18,43,44,32)(24,39,52,66)(27,51,56,42)(28,58,57,41)(31,46,62,64)(33,45,63,61)(49,68,69,54)(53,59,65,72)(55,70,71,67);(2,5)(6,8)(9,11,14,27,15,28,35,38,16,25,36,56,34,57,13,26)(10,17,20,41,21,42,47,50,22,40,48,58,37,51,19,23)(12,24,32,54,33,55,62,65,30,52,43,68,63,71,31,53)(18,39,29,59,46,67,61,69,44,66,60,72,64,70,45,49);(1,3,4,7)(2,6,5,8)(9,12,13,31,34,63,36,43,16,30,35,62,15,33,14,32)(10,18,19,45,37,64,48,60,22,44,47,61,21,46,20,29)(11,24,26,53,57,71,56,68,25,52,38,65,28,55,27,54)(17,39,23,49,51,70,58,72,40,66,50,69,42,67,41,59);(12,30)(18,44)(24,52)(29,60)(31,62)(32,43)(33,63)(39,66)(45,61)(46,64)(49,69)(53,65)(54,68)(55,71)(59,72)(67,70);(11,25)(17,40)(23,50)(24,52)(26,38)(27,56)(28,57)(39,66)(41,58)(42,51)(49,69)(53,65)(54,68)(55,71)(59,72)(67,70)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=f,f^2=hi,g^2=h,h^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=ghi,h^a=hi,e^b=ei,d^c=di

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7530371);