#8096114

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

4967 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=g,d^2=g,e^2=h,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, 8003310);