#9684085

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

4509 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=hi,d^2=g,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,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, 9591281);