#8254862

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

5324 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)(11,16)(12,14)(13,15)(17,18,23,24)(19,25,20,29)(21,22,26,32)(27,31,30,28);(1,2,5,4)(3,7,8,6)(9,11,14,13)(10,15,12,16)(18,24)(21,26)(25,29)(27,30);(2,4)(6,7)(9,10,14,12)(11,15,13,16)(17,19,23,20)(18,25,24,29)(21,27,26,30)(22,31,32,28);(9,10,14,12)(11,16,13,15)(17,20,23,19)(18,25,24,29)(21,30,26,27)(22,31,32,28);(1,3,5,8)(2,6,4,7)(9,12,14,10)(11,15,13,16)(17,21,23,26)(18,22,24,32)(19,27,20,30)(25,31,29,28);(1,4,5,2)(3,7,8,6)(9,13,14,11)(10,15,12,16)(17,22)(18,26)(19,28)(20,31)(21,24)(23,32)(25,27)(29,30)

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=g,d^2=g,e^2=gi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi

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, 8162058);