#383617

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

1065 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)(11,20)(12,17,26,16)(13,18)(14,19)(15,23)(21,31,40,39)(22,35,25,32)(24,38)(27,36,30,34)(28,37)(29,33);(1,2,4,5)(3,8,7,6)(9,11,13,23)(10,15,18,20)(12,25,26,22)(14,29,28,24)(16,32,17,35)(19,38,37,33)(21,30,40,27)(31,36,39,34);(9,12)(10,16)(11,21)(13,26)(14,30)(15,31)(17,18)(19,36)(20,39)(22,24)(23,40)(25,29)(27,28)(32,38)(33,35)(34,37);(1,3,4,7)(2,6,5,8)(9,12)(10,17)(11,22)(13,26)(14,27)(15,32)(16,18)(19,36)(20,35)(21,24)(23,25)(28,30)(29,40)(31,38)(33,39)(34,37)

Polycyclic presentation:

2^9|a^2=h,b^2=e,d^2=ei,g^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, 290813);