#371627

order	512
nilpotency class	3
isoclinism	isoclinic class of #24802
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₄×#121
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

873 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=fh,c^2=e,d^2=eg,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,f^c=fi,h^c=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 278823);