#8458818

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

4648 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,e^2=hi,f^2=gi,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, 8366014);