#8093255

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

4808 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|c^2=hi,e^2=g,f^2=h,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, 8000451);