#8096211

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

4949 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:

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

Polycyclic presentation:

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

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, 8003407);