#8461280

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5021 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|b^2=gi,c^2=gh,d^2=g,e^2=hi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8368476);