#10088402

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

4560 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=gh,d^2=gh,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9995598);