#8496608

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

4852 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:

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

Polycyclic presentation:

2^9|a^2=g,d^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=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, 8403804);