#399110

order	512
nilpotency class	3
isoclinism	isoclinic class of #393906
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

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×#532	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1284 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:

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

Polycyclic presentation:

2^9|b^2=ef,c^2=ef,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,h^b=hi,e^c=ei,f^c=fi,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, 306306);