#8399321

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

4921 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)(11,15)(12,14)(13,16)(17,18,24,23)(19,21,25,20)(22,30,32,26)(27,28,31,29);(1,2)(3,4)(5,6)(7,8)(9,10,12,14)(11,13,16,15)(17,19)(18,21)(20,23)(22,31)(24,25)(26,28)(27,32)(29,30);(2,6)(4,7)(9,10)(11,13)(12,14)(15,16)(17,20,24,21)(18,25,23,19)(22,28,32,29)(26,31,30,27);(2,6)(3,8)(9,11,12,16)(10,13,14,15)(17,21,24,20)(18,25,23,19)(22,28,32,29)(26,27,30,31);(1,3)(2,7)(4,6)(5,8)(17,22,24,32)(18,26,23,30)(19,27,25,31)(20,28,21,29);(1,4,5,7)(2,3,6,8)(17,23,24,18)(19,20,25,21)(22,30,32,26)(27,28,31,29)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=h,d^2=gh,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8306517);