#456557

order	512
nilpotency class	3
isoclinism	isoclinic class of #453948
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#537	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1411 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=efi,c^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,g^b=gi,d^c=de,g^c=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, 363753);