#8093301

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

5308 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)(3,8)(4,7)(5,6)(9,10)(11,12)(13,14)(15,16)(18,23)(21,31)(24,30)(25,32);(2,6)(7,8)(10,14)(11,16)(19,27)(20,29)(24,30)(25,32);(9,11,13,16)(10,15,14,12)(17,18,22,23)(19,24,27,30)(20,25,29,32)(21,26,31,28);(1,3,5,4)(2,7,6,8)(9,11,13,16)(10,15,14,12)(17,19)(18,24)(20,28)(21,25)(22,27)(23,30)(26,29)(31,32);(1,2)(3,7)(4,8)(5,6)(9,10)(11,15)(12,16)(13,14)(17,20,22,29)(18,25,23,32)(19,28,27,26)(21,30,31,24);(1,4,5,3)(2,8,6,7)(9,12,13,15)(10,16,14,11)(17,21)(18,26)(19,25)(20,30)(22,31)(23,28)(24,29)(27,32)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8000497);