#371001

order	512
nilpotency class	3
isoclinism	isoclinic class of #23876
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₄×#120
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1270 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=eh,d^2=g,e^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,e^b=ei,f^b=fi,h^b=hi,e^c=ei,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, 278197);