#10372311

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

4687 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10279507);