#8093652

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

5209 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:

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

Polycyclic presentation:

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

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, 8000848);