#392548

order	512
nilpotency class	2
isoclinism	isoclinic class of #14715
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂³×C₄	C₂⁴
commutator, socle, prime radical	C₂⁴	C₂³×C₄
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

990 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fg,c^2=ef,d^2=e,h^2=i,b^a=be,c^a=cf,d^a=dg,c^b=ci

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 299744);