#8108128

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

4898 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)(18,23)(19,27)(20,29)(21,31);(2,5)(6,8)(11,15)(14,16)(17,18,22,23)(19,24,27,30)(20,28,29,25)(21,26,31,32);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19,22,27)(18,24,23,30)(20,26,29,32)(21,28,31,25);(9,11,12,15)(10,14,13,16)(17,20,22,29)(18,25,23,28)(19,26,27,32)(21,30,31,24);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,24)(20,25)(21,26)(22,23)(27,30)(28,29)(31,32);(1,3)(2,6)(4,7)(5,8)(17,21)(18,26)(19,28)(20,30)(22,31)(23,32)(24,29)(25,27)

Polycyclic presentation:

2^9|b^2=g,c^2=g,d^2=h,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, 8015324);