#8095856

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

4879 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8003052);