#8492753

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

4346 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=i,d^2=g,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,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, 8399949);