#8750819

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

4721 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)(3,5)(4,7)(6,8)(11,12)(14,15)(17,18,23,22)(19,26,21,24)(20,25,28,30)(27,32,29,31);(2,3)(6,7)(9,10,13,16)(11,14,12,15)(17,19,23,21)(18,24,22,26)(20,27,28,29)(25,31,30,32);(1,2,5,3)(4,6,8,7)(9,11,13,12)(10,14,16,15)(20,28)(25,30)(27,29)(31,32);(1,3,5,2)(4,7,8,6)(9,10)(11,14)(12,15)(13,16)(17,20)(18,25)(19,27)(21,29)(22,30)(23,28)(24,31)(26,32);(1,4)(2,6)(3,7)(5,8)(9,12,13,11)(10,15,16,14)(17,21,23,19)(18,24,22,26)(20,29,28,27)(25,31,30,32);(17,22,23,18)(19,26,21,24)(20,25,28,30)(27,31,29,32)

Polycyclic presentation:

2^9|a^2=gi,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, 8658015);