#9684086

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

5003 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,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, 9591282);