#8193650

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	C₂³×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4472 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)(3,7)(4,5)(6,8)(25,26,32,35)(27,29,37,28)(30,33,36,39)(31,34,40,38);(1,2)(3,6)(4,5)(7,8)(9,10)(11,17)(12,16)(13,14)(15,19)(18,23)(20,21)(22,24)(25,27,32,37)(26,28,35,29)(30,31,36,40)(33,38,39,34);(2,5)(6,7)(10,14)(15,17)(16,21)(22,23)(25,28,32,29)(26,27,35,37)(30,38,36,34)(31,39,40,33);(9,11,13,19)(10,15,14,17)(12,18,20,24)(16,22,21,23)(25,29,32,28)(26,27,35,37)(30,34,36,38)(31,39,40,33);(1,3)(2,6)(4,8)(5,7)(9,12,13,20)(10,16,14,21)(11,18,19,24)(15,22,17,23)(25,30,32,36)(26,33,35,39)(27,31,37,40)(28,38,29,34);(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,21)(23,24)(25,31)(26,34)(27,36)(28,33)(29,39)(30,37)(32,40)(35,38)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=g,d^2=ghi,e^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8100846);