#10086503

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

4942 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=g,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=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, 9993699);