#10451641

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

4325 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=gh,d^2=gi,e^2=hi,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,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, 10358837);