#8105987

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

5866 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8013183);