#401111

order	512
nilpotency class	3
isoclinism	isoclinic class of #393906
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2408
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1275 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 308307);