#8094938

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

4677 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,d^2=g,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^c=eghi

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, 8002134);