#8095839

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

5120 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)(11,12)(13,14)(15,16)(17,18)(19,22)(20,27)(21,29)(23,25)(24,32)(26,28)(30,31);(1,2)(3,6)(4,7)(5,8)(10,14)(11,15)(17,19)(18,20)(21,30)(22,25)(23,27)(24,31)(26,32)(28,29);(1,3)(2,5)(4,8)(6,7)(9,11,13,15)(10,12,14,16)(17,20,23,22)(18,19,25,27)(21,28,32,31)(24,26,29,30);(9,12)(10,15)(11,14)(13,16)(17,21)(18,24)(19,26)(20,28)(22,31)(23,32)(25,29)(27,30);(1,3)(2,6)(4,8)(5,7)(17,22,23,20)(18,19,25,27)(21,31,32,28)(24,26,29,30);(1,2)(3,5)(4,7)(6,8)(17,19)(18,22)(20,25)(21,26)(23,27)(24,31)(28,29)(30,32)

Polycyclic presentation:

2^9|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, 8003035);