#8455755

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

4751 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)(10,15)(11,17)(12,19)(20,24)(25,26,31,35)(27,30,34,39)(28,38,37,32)(29,33,36,40);(3,7)(6,8)(9,10,14,15)(11,13,17,21)(12,16,19,22)(18,20,23,24)(25,27,31,34)(26,30,35,39)(28,29,37,36)(32,40,38,33);(1,2)(3,6)(4,5)(7,8)(25,28,31,37)(26,32,35,38)(27,36,34,29)(30,33,39,40);(1,3)(2,6)(4,7)(5,8)(9,11)(10,13)(12,18)(14,17)(15,21)(16,20)(19,23)(22,24)(25,27)(26,30)(28,36)(29,37)(31,34)(32,33)(35,39)(38,40);(1,2)(3,6)(4,5)(7,8)(9,12)(10,16)(11,18)(13,20)(14,19)(15,22)(17,23)(21,24)(25,29,31,36)(26,33,35,40)(27,37,34,28)(30,32,39,38);(9,13)(10,17)(11,15)(12,20)(14,21)(16,23)(18,22)(19,24)(25,30)(26,34)(27,35)(28,33)(29,32)(31,39)(36,38)(37,40)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=i,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, 8362951);