#9712802

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

4561 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,d^2=h,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,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, 9619998);