#452154

order	512
nilpotency class	3
isoclinism	isoclinic class of #417541
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2404
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1175 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:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=f,d^2=f,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,d^b=dei,e^c=ei,g^d=gi,h^d=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 359350);