#475112

order	512
nilpotency class	3
isoclinism	isoclinic class of #454974
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2598
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

835 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=eh,d^2=g,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^c=de,e^c=ei,f^c=fi,g^c=gi,f^d=fi

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, 382308);