#8095854

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

5144 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)(17,18,23,26)(19,21,28,24)(20,29,22,25)(27,32,30,31);(1,2)(3,6)(4,7)(5,8)(10,11)(14,15)(17,19)(18,24)(20,30)(21,26)(22,27)(23,28)(25,31)(29,32);(1,3)(2,5)(4,8)(6,7)(9,11,13,10)(12,15,16,14)(17,20,23,22)(18,25,26,29)(19,27,28,30)(21,31,24,32);(9,12)(10,14)(11,15)(13,16)(17,21)(18,19)(20,31)(22,32)(23,24)(25,27)(26,28)(29,30);(1,3)(2,6)(4,8)(5,7)(17,22,23,20)(18,25,26,29)(19,27,28,30)(21,32,24,31);(1,2)(3,5)(4,7)(6,8)(17,19)(18,21)(20,27)(22,30)(23,28)(24,26)(25,31)(29,32)

Polycyclic presentation:

2^9|a^2=g,c^2=g,e^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, 8003050);