#10363630

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

4320 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)(9,10,15,18)(11,16,12,14)(13,17,22,24)(19,23,20,21)(25,26,29,30)(27,33,35,31)(28,32,37,38)(34,40,36,39);(11,12)(14,16)(19,20)(21,23)(26,30)(31,33)(32,38)(39,40);(1,2)(3,6)(4,5)(7,8)(9,11,15,12)(10,16,18,14)(13,20,22,19)(17,21,24,23)(25,27,29,35)(26,31,30,33)(28,36,37,34)(32,40,38,39);(1,3,4,7)(2,6,5,8)(9,12,15,11)(10,14,18,16)(13,19,22,20)(17,23,24,21)(25,26)(27,31)(28,32)(29,30)(33,35)(34,39)(36,40)(37,38);(1,3)(2,6)(4,7)(5,8)(9,13)(10,17)(11,19)(12,20)(14,21)(15,22)(16,23)(18,24)(25,28,29,37)(26,32,30,38)(27,34,35,36)(31,39,33,40);(1,2)(3,6)(4,5)(7,8)(9,14)(10,11)(12,18)(13,21)(15,16)(17,19)(20,24)(22,23)(25,26)(27,31)(28,32)(29,30)(33,35)(34,39)(36,40)(37,38)

Polycyclic presentation:

2^9|a^2=gi,c^2=gi,d^2=ghi,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi

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, 10270826);