#394291

order	512
nilpotency class	3
isoclinism	isoclinic class of #394289
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2632 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=ei,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,h^d=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 301487);