#8968576

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

4864 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)(11,13)(12,16)(14,15)(18,23)(19,25)(24,30)(26,31);(1,2)(3,4)(5,7)(6,8)(10,15)(11,16)(17,18)(19,20)(21,24)(22,23)(25,27)(26,28)(29,30)(31,32);(1,3)(2,6)(4,7)(5,8)(9,11,14,16)(10,12,15,13)(19,25)(20,27)(21,29)(24,30);(3,8)(4,6)(9,12,14,13)(10,11,15,16)(17,19,22,25)(18,20,23,27)(21,26,29,31)(24,28,30,32);(1,4)(2,3)(5,6)(7,8)(9,10)(11,12)(13,16)(14,15)(17,20)(18,19)(21,28)(22,27)(23,25)(24,26)(29,32)(30,31);(1,2)(3,4)(5,7)(6,8)(9,13,14,12)(10,16,15,11)(17,21,22,29)(18,24,23,30)(19,26,25,31)(20,28,27,32)

Polycyclic presentation:

2^9|c^2=h,d^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=dgi,e^c=egi,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, 8875772);