#10106991

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

4337 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)(3,4)(5,8)(6,7)(9,10)(11,19)(12,17)(13,21)(14,15)(16,20)(18,24)(22,23)(25,26,31,35)(27,30,34,32)(28,29,33,40)(36,38,39,37);(1,3,5,7)(2,4,8,6)(10,15)(12,23)(16,19)(18,21)(25,27,31,34)(26,32,35,30)(28,36,33,39)(29,37,40,38);(1,3)(2,6)(4,8)(5,7)(10,15)(11,20)(17,22)(18,21)(25,28,31,33)(26,29,35,40)(27,36,34,39)(30,38,32,37);(1,3,5,7)(2,4,8,6)(9,11,14,20)(10,16,15,19)(12,18,23,21)(13,22,24,17)(25,26)(27,32)(28,29)(30,34)(31,35)(33,40)(36,37)(38,39);(1,3)(2,4)(5,7)(6,8)(9,12)(10,17)(11,21)(13,19)(14,23)(15,22)(16,24)(18,20)(25,29)(26,33)(27,37)(28,35)(30,36)(31,40)(32,39)(34,38);(1,4)(2,7)(3,8)(5,6)(9,13,14,24)(10,18,15,21)(11,22,20,17)(12,16,23,19)(25,30,31,32)(26,34,35,27)(28,38,33,37)(29,39,40,36)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=ghi,d^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, 10014187);