#8750818

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

4600 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=h,d^2=i,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8658014);