#418502

order	512
nilpotency class	3
isoclinism	isoclinic class of #418500
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

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

1683 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)(3,4)(9,10,15,20)(11,21,24,35)(12,25,28,17)(13,19,14,18)(16,29,32,23)(22,33,38,39)(26,31,37,40)(27,34,30,36);(5,6)(7,8)(9,11)(10,16)(12,26)(13,29)(14,23)(15,24)(17,33)(18,35)(19,21)(20,32)(22,27)(25,39)(28,37)(30,38)(31,34)(36,40);(2,4)(6,8)(9,11)(10,16)(12,22)(13,23)(14,29)(15,24)(17,31)(18,21)(19,35)(20,32)(25,40)(26,27)(28,38)(30,37)(33,34)(36,39);(9,12,13,27,15,28,14,30)(10,17,18,34,20,25,19,36)(11,22,23,26,24,38,29,37)(16,31,21,33,32,40,35,39)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 325698);