#9064899

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

5033 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)(11,15)(12,16)(13,14)(17,18,22,25)(19,20,27,23)(21,24,29,31)(26,28,32,30);(1,2)(3,4)(5,6)(7,8)(9,10,13,14)(11,15,16,12)(17,19)(18,23)(20,25)(21,26)(22,27)(24,30)(28,31)(29,32);(2,6)(3,7)(9,11)(10,12)(13,16)(14,15)(21,29)(24,31)(26,32)(28,30);(3,7)(4,8)(9,12)(10,11)(13,15)(14,16)(17,20,22,23)(18,19,25,27)(21,30,29,28)(24,32,31,26);(1,3)(2,4)(5,7)(6,8)(9,10)(11,12)(13,14)(15,16)(17,21,22,29)(18,24,25,31)(19,26,27,32)(20,28,23,30);(1,4)(2,3)(5,8)(6,7)(9,12)(10,11)(13,15)(14,16)(17,19)(18,20)(21,26)(22,27)(23,25)(24,28)(29,32)(30,31)

Polycyclic presentation:

2^9|a^2=gh,b^2=h,d^2=gh,e^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,e^d=egi,f^d=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, 8972095);