#8201600

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

5040 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8108796);