#8445782

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

4532 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)(6,8)(9,10,15,17)(11,18,22,16)(12,14,13,23)(19,24,20,21)(25,26,32,36)(27,33,30,31)(28,38,37,34)(29,35,39,40);(3,7)(6,8)(9,11)(10,16)(12,19)(13,20)(14,21)(15,22)(17,18)(23,24)(25,27,32,30)(26,33,36,31)(28,39,37,29)(34,35,38,40);(1,2)(3,6)(4,5)(7,8)(9,12,15,13)(10,14,17,23)(11,19,22,20)(16,21,18,24)(25,28)(26,34)(27,29)(30,39)(31,35)(32,37)(33,40)(36,38);(1,3,4,7)(2,6,5,8)(9,13,15,12)(10,14,17,23)(11,20,22,19)(16,21,18,24)(25,29,32,39)(26,35,36,40)(27,37,30,28)(31,38,33,34);(1,2)(3,6)(4,5)(7,8)(9,11)(10,16)(12,19)(13,20)(14,21)(15,22)(17,18)(23,24)(25,30,32,27)(26,31,36,33)(28,39,37,29)(34,35,38,40);(9,14)(10,13)(11,21)(12,17)(15,23)(16,20)(18,19)(22,24)(25,31)(26,27)(28,35)(29,34)(30,36)(32,33)(37,40)(38,39)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=g,d^2=gh,e^2=i,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, 8352978);