#6825552

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806839
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#24790	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1534 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,24)(20,31)(21,29)(22,27)(23,25)(26,28)(30,32);(2,5)(3,7)(17,18,22,27)(19,28,30,21)(20,25,23,31)(24,29,32,26);(10,13)(14,16)(17,19,23,26,22,30,20,29)(18,24,25,28,27,32,31,21);(2,5)(6,8)(9,11,12,15)(10,14,13,16)(17,20,22,23)(18,25,27,31)(19,29,30,26)(21,24,28,32);(1,3,4,7)(2,6,5,8)(17,21,23,24,22,28,20,32)(18,26,25,30,27,29,31,19)

Polycyclic presentation:

2^9|b^2=i,c^2=h,d^2=gi,e^2=fhi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6732748);