#383400

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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1098 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)(3,5)(4,6)(7,8)(9,10,14,20,15,17,13,19)(11,16,23,29,24,31,21,33)(12,25,26,34,27,18,28,32)(22,36,35,39,38,30,37,40);(1,3,5,2)(4,7,8,6)(9,11)(10,16)(12,22)(13,21)(14,23)(15,24)(17,31)(18,30)(19,33)(20,29)(25,36)(26,35)(27,38)(28,37)(32,40)(34,39);(1,4,5,8)(2,6,3,7)(10,17)(11,21)(13,14)(16,29)(18,25)(22,35)(23,24)(26,28)(30,39)(31,33)(36,40)(37,38);(2,3)(6,7)(9,12,15,27)(10,18,17,25)(11,22,24,38)(13,26,14,28)(16,30,31,36)(19,32,20,34)(21,35,23,37)(29,39,33,40)

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=e,d^2=i,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, 290596);