#8108311

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

4996 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)(11,16)(12,14)(13,15)(17,18)(19,24)(20,29)(21,31)(22,23)(25,30)(26,27)(28,32);(2,5)(6,8)(10,12)(15,16)(18,23)(24,32)(25,29)(26,31);(1,2)(3,6)(4,5)(7,8)(9,11,14,13)(10,15,12,16)(17,19,22,28)(18,24,23,32)(20,27,30,21)(25,31,29,26);(9,10,14,12)(11,15,13,16)(17,20)(18,25)(19,27)(21,28)(22,30)(23,29)(24,31)(26,32);(1,2)(3,6)(4,5)(7,8)(9,12,14,10)(11,16,13,15)(17,18)(19,24)(20,25)(21,26)(22,23)(27,31)(28,32)(29,30);(1,3)(2,6)(4,7)(5,8)(9,13,14,11)(10,16,12,15)(17,21,22,27)(18,26,23,31)(19,20,28,30)(24,25,32,29)

Polycyclic presentation:

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

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, 8015507);