#8463607

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

4732 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)(4,6)(5,7)(9,10,13,14)(11,15,16,12)(17,18,22,23)(19,20,25,28)(21,24,29,30)(26,27,31,32);(1,3,5,4)(2,6,7,8)(9,11,13,16)(10,12,14,15)(18,23)(19,25)(24,30)(26,31);(9,10)(11,12)(13,14)(15,16)(17,19)(18,20)(21,26)(22,25)(23,28)(24,27)(29,31)(30,32);(2,7)(6,8)(9,10)(11,15)(12,16)(13,14)(17,20)(18,19)(21,27)(22,28)(23,25)(24,26)(29,32)(30,31);(1,4,5,3)(2,8,7,6);(1,2)(3,6)(4,8)(5,7)(9,12,13,15)(10,11,14,16)(17,21,22,29)(18,24,23,30)(19,26,25,31)(20,27,28,32)

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,e^2=ghi,f^2=hi,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, 8370803);