#8437811

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

4388 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)(3,7)(4,5)(6,8)(9,10,13,14)(11,17,19,15)(12,16,20,21)(18,23,24,22)(25,26,32,31)(27,33,30,35)(28,38,29,34)(36,40,37,39);(2,5)(6,7)(10,14)(15,17)(16,21)(22,23)(25,27,32,30)(26,33,31,35)(28,37,29,36)(34,40,38,39);(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,21)(23,24)(25,28,32,29)(26,34,31,38)(27,36,30,37)(33,39,35,40);(1,3,4,8)(2,6,5,7)(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,21)(23,24)(25,29,32,28)(26,34,31,38)(27,37,30,36)(33,39,35,40);(9,11,13,19)(10,15,14,17)(12,18,20,24)(16,22,21,23)(25,30,32,27)(26,35,31,33)(28,37,29,36)(34,40,38,39);(9,12,13,20)(10,16,14,21)(11,18,19,24)(15,22,17,23)(25,31,32,26)(27,35,30,33)(28,38,29,34)(36,40,37,39)

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=gi,d^2=gh,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8345007);