#370819

order	512
nilpotency class	3
isoclinism	isoclinic class of #25140
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	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

784 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)(3,7)(4,5)(6,8)(9,10)(11,15,21,31)(12,17,23,34)(13,19)(14,18)(16,22,30,25)(20,36)(24,33,26,35)(27,28)(29,38)(32,39)(37,40);(2,5)(3,8)(9,11)(10,15,18,31)(12,20)(13,25)(14,21)(16,28,30,19)(17,29,34,36)(22,27)(23,38)(24,39)(26,37)(32,33,40,35);(1,3,4,8)(2,6,5,7)(9,11)(10,16)(12,20)(13,25)(14,21)(15,28)(17,32)(18,30)(19,31)(22,27)(23,38)(24,39)(26,37)(29,33)(34,40)(35,36);(2,5)(6,7)(9,12,13,26,14,23,27,24)(10,17,28,33,18,34,19,35)(11,20,25,37,21,38,22,39)(15,29,30,40,31,36,16,32)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 278015);