#8505644

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

4488 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=hi,d^2=gi,e^2=h,f^2=hi,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, 8412840);