#480304

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	C₂²×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×D₄ ◁ C₂×#520 ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

1018 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=i,e^2=i,f^2=h,b^a=be,c^a=ch,e^a=eg,g^a=gi,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, 387500);