#8105967

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

4991 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)(10,13)(14,16)(17,18,22,24)(19,27,29,23)(20,30,31,25)(21,32,28,26);(1,2)(3,6)(4,5)(7,8)(11,15)(14,16)(17,19,22,29)(18,23,24,27)(20,28,31,21)(25,26,30,32);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(18,24)(23,27)(25,30)(26,32);(1,2)(3,6)(4,5)(7,8)(9,11)(10,14)(12,15)(13,16)(17,18)(19,27)(20,25)(21,26)(22,24)(23,29)(28,32)(30,31);(9,10)(11,14)(12,13)(15,16)(17,20)(18,25)(19,28)(21,29)(22,31)(23,26)(24,30)(27,32);(1,3)(2,6)(4,7)(5,8)(17,21,22,28)(18,26,24,32)(19,20,29,31)(23,30,27,25)

Polycyclic presentation:

2^9|a^2=g,b^2=g,f^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, 8013163);