#10361125

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

4627 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=gi,d^2=hi,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, 10268321);