#8533692

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

4688 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=i,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=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, 8440888);