#8095106

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
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4237 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

(1,2,5,6)(3,7,8,4)(10,14)(11,17)(16,21)(18,23)(28,36)(32,37)(34,40)(38,39);(3,8)(4,7)(9,10)(11,15)(12,16)(13,14)(17,20)(18,22)(19,21)(23,24)(26,30)(31,33)(32,37)(38,39);(1,3)(2,4)(5,8)(6,7)(12,19)(16,21)(18,23)(22,24)(25,26,29,30)(27,33,35,31)(28,32,36,37)(34,39,40,38);(1,4)(2,3)(5,7)(6,8)(9,11,13,17)(10,15,14,20)(12,18,19,23)(16,22,21,24);(9,12,13,19)(10,16,14,21)(11,18,17,23)(15,22,20,24)(25,27,29,35)(26,31,30,33)(28,34,36,40)(32,38,37,39);(9,10)(11,15)(12,16)(13,14)(17,20)(18,22)(19,21)(23,24)(25,28)(26,32)(27,34)(29,36)(30,37)(31,38)(33,39)(35,40)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8002302);