#8799162

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

4827 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=h,d^2=gi,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8706358);