#9713344

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

4833 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=h,d^2=h,e^2=gi,f^2=ghi,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, 9620540);