#8323181

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

4979 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,d^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=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, 8230377);