#8612870

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

5652 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,d^2=g,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=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, 8520066);