#9123527

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

5358 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=g,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9030723);