#8199213

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

4718 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8106409);