#6812265

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806903
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	#39934	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#120 ◁ 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=h,d^2=i,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,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, 6719461);