#10202057

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

4551 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:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=h,d^2=ghi,e^2=gh,f^2=ghi,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, 10109253);