#408253

order	512
nilpotency class	3
isoclinism	isoclinic class of #393989
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#127
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2397
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1049 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)(11,16)(12,23,25,36)(13,18)(14,19)(15,20)(17,26,34,28)(21,38,39,33)(22,31)(24,30,40,32)(27,35)(29,37);(1,3,4,7)(2,5,6,8)(9,11,14,13,15,22,29,27)(10,16,19,18,20,31,37,35)(12,21,28,24,25,39,26,40)(17,32,36,38,34,30,23,33);(9,12)(10,17)(11,21)(13,24)(14,28)(15,25)(16,30)(18,33)(19,36)(20,34)(22,39)(23,37)(26,29)(27,40)(31,32)(35,38);(2,6)(5,8)(9,13,14,22,15,27,29,11)(10,18,19,31,20,35,37,16)(12,24,28,39,25,40,26,21)(17,33,36,32,34,38,23,30)

Polycyclic presentation:

2^9|a^2=e,b^2=efh,d^2=ef,f^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=di,e^c=ei,f^c=fi

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, 315449);