#8461241

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

4500 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,d^2=g,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, 8368437);