#10068918

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

5196 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,d^2=i,e^2=i,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi

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, 9976114);