#8253918

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

4580 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,d^2=g,e^2=gi,f^2=i,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, 8161114);