#383825

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

1013 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)(3,6)(4,7)(5,8)(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);(1,3,6,2)(4,8,5,7)(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);(1,4,6,5)(2,7,3,8)(10,17)(11,21)(13,14)(16,29)(18,25)(22,35)(23,24)(26,28)(30,39)(31,33)(36,40)(37,38);(1,5,6,4)(2,7,3,8)(9,12)(10,18)(11,22)(13,26)(14,28)(15,27)(16,30)(17,25)(19,32)(20,34)(21,35)(23,37)(24,38)(29,39)(31,36)(33,40)

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=e,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, 291021);