#9888134

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

5185 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=gi,d^2=g,e^2=g,f^2=i,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,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, 9795330);