#383619

order	512
nilpotency class	3
isoclinism	isoclinic class of #222272
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1016 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=i,d^2=e,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,c^b=cg,d^b=de,g^b=gi,g^c=gi,h^c=hi,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, 290815);