#8463430

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

4785 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=h,d^2=g,e^2=i,f^2=hi,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, 8370626);