#8750804

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

4579 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:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=h,d^2=i,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8658000);