#9203714

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

5010 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)(3,7,5,4)(9,10)(11,12)(13,14)(15,16)(17,18)(19,23)(20,21)(22,25)(24,26)(27,28)(29,32)(30,31);(1,3,6,5)(2,4,8,7)(10,14)(11,15)(17,19)(18,23)(20,27)(21,28)(22,26)(24,25)(29,30)(31,32);(1,4)(2,5)(3,8)(6,7)(9,11,13,15)(10,12,14,16)(19,26)(23,24)(27,29)(28,32);(1,4,6,7)(2,3,8,5)(9,12)(10,15)(11,14)(13,16)(17,19)(18,24)(20,29)(21,28)(22,26)(23,25)(27,30)(31,32);(1,4)(2,3)(5,8)(6,7)(17,20,22,30)(18,21,25,31)(19,27,26,29)(23,28,24,32);(1,5,6,3)(2,7,8,4)(17,21)(18,20)(19,28)(22,31)(23,27)(24,29)(25,30)(26,32)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=gh,d^2=h,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=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, 9110910);