#9683886

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

4640 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)(11,16)(12,15)(13,14)(17,18,24,29)(19,27,21,25)(20,26,31,30)(22,23,28,32);(1,2)(3,8)(4,6)(5,7)(9,10,14,13)(11,15,12,16)(17,19,24,21)(18,25,29,27)(20,28,31,22)(23,30,32,26);(1,3,5,6)(2,4,7,8)(9,10,14,13)(11,16,12,15)(17,20,24,31)(18,26,29,30)(19,28,21,22)(23,27,32,25);(9,11,14,12)(10,15,13,16)(17,21,24,19)(18,27,29,25)(20,22,31,28)(23,26,32,30);(1,2)(3,4)(5,7)(6,8)(9,12,14,11)(10,15,13,16)(17,22)(18,23)(19,20)(21,31)(24,28)(25,30)(26,27)(29,32);(1,4)(2,6)(3,7)(5,8)(9,13,14,10)(11,16,12,15)(17,23)(18,28)(19,30)(20,27)(21,26)(22,29)(24,32)(25,31)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=gi,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, 9591082);