#9127605

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

4753 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=h,e^2=h,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, 9034801);