#480429

order	512
nilpotency class	4
isoclinism	isoclinic class of #480274
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄⋊C₄	C₂⁴
prime radical	C₂×#532	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₂²⋊C₄ ◁ C₂×#520 ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

1030 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:

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

Polycyclic presentation:

2^9|a^2=f,d^2=hi,e^2=i,f^2=h,b^a=be,c^a=ch,e^a=eg,g^a=gi,c^b=ci,e^b=ei,f^b=fgi,d^c=di,f^e=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, 387625);