#8093174

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

5896 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:

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

Polycyclic presentation:

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

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, 8000370);