#8096167

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

5557 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)(3,7)(4,8)(5,6)(9,10)(11,15)(12,14)(13,16)(17,18)(19,27)(20,22)(21,24)(23,26)(25,31)(28,32)(29,30);(2,6)(4,7)(9,10,12,14)(11,13,16,15)(17,19,23,21)(18,24,26,27)(20,28,31,30)(22,29,25,32);(1,3)(2,4)(5,8)(6,7)(9,10)(11,13)(12,14)(15,16);(1,4)(2,3)(5,7)(6,8)(9,11)(10,13)(12,16)(14,15)(17,20)(18,25)(19,28)(21,30)(22,26)(23,31)(24,32)(27,29);(17,21,23,19)(18,24,26,27)(20,30,31,28)(22,29,25,32);(17,22,23,25)(18,20,26,31)(19,29,21,32)(24,28,27,30)

Polycyclic presentation:

2^9|b^2=gh,e^2=ghi,f^2=ghi,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, 8003363);