#10324079

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

4276 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=g,d^2=gh,e^2=h,f^2=gh,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, 10231275);