#8471132

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

4961 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=hi,d^2=i,e^2=gi,f^2=ghi,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, 8378328);