#9715486

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

4958 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)(3,8)(4,6)(5,7)(9,10)(11,15)(12,14)(13,16)(17,18,22,24)(19,23,26,20)(21,29,31,25)(27,28,32,30);(1,3,5,6)(2,4,7,8)(9,10,12,14)(11,13,16,15)(17,19)(18,23)(20,24)(21,27)(22,26)(25,30)(28,29)(31,32);(9,10)(11,13)(12,14)(15,16)(18,24)(19,26)(25,29)(27,32);(9,11)(10,13)(12,16)(14,15)(17,19)(18,20)(21,27)(22,26)(23,24)(25,28)(29,30)(31,32);(1,2)(3,4)(5,7)(6,8)(17,20)(18,19)(21,30)(22,23)(24,26)(25,32)(27,29)(28,31);(1,4)(2,6)(3,7)(5,8)(17,21,22,31)(18,25,24,29)(19,27,26,32)(20,28,23,30)

Polycyclic presentation:

2^9|a^2=h,b^2=g,f^2=h,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, 9622682);