#8461251

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

5441 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:

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

Polycyclic presentation:

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

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, 8368447);