#480251

order	512
nilpotency class	4
isoclinism	isoclinic class of #480250
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂×C₂²⋊C₄	C₂⁴
prime radical	C₂³×#41	C₄
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

1618 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)(12,16)(13,25,26,17)(14,18)(15,22,28,35)(20,31)(21,32)(23,36,37,29)(24,30)(27,34,38,40)(33,39);(9,11)(10,15)(12,20)(13,22)(14,21)(16,27)(17,19)(18,28)(23,34)(24,33)(25,32)(26,35)(29,31)(30,38)(36,39)(37,40);(2,5)(6,8)(9,12,14,24)(10,16,18,30)(11,20,21,33)(13,23,26,37)(15,27,28,38)(17,29,25,36)(19,31,32,39)(22,34,35,40);(11,21)(15,28)(19,32)(20,33)(22,35)(27,38)(31,39)(34,40)

Polycyclic presentation:

2^9|a^2=f,c^2=i,f^2=h,b^a=be,c^a=ch,e^a=eg,g^a=gi,d^b=di,f^b=fg,f^e=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72284215, #72284218, #72284224, …
maximal normal subgroup of	#72284215, #72284218, #72284224, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 387447);