#358781

order	512
nilpotency class	3
isoclinism	isoclinic class of #8325
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#379
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#379
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2670 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 265977);