#381250

order	512
nilpotency class	3
isoclinism	isoclinic class of #214664
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂×#534	C₂²
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1441 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)(11,19,22,15)(12,16)(13,17)(14,18)(20,26,35,28)(21,33,36,27)(23,29)(24,30)(25,32)(31,37)(34,38,40,39);(2,5)(6,8)(9,11)(10,15)(12,20)(13,21)(14,22)(16,28)(17,27)(18,19)(23,34)(24,35)(25,36)(26,30)(29,39)(31,38)(32,33)(37,40);(3,7)(6,8)(9,12,14,24)(10,16,18,30)(11,20,22,35)(13,23,25,37)(15,26,19,28)(17,31,32,29)(21,34,36,40)(27,39,33,38);(1,3,4,7)(2,6,5,8)(9,13)(10,17)(11,21)(12,23)(14,25)(15,27)(16,29)(18,32)(19,33)(20,34)(22,36)(24,37)(26,38)(28,39)(30,31)(35,40)

Polycyclic presentation:

2^9|a^2=h,c^2=i,d^2=eg,b^a=be,c^a=cf,e^a=ei,g^a=gi,c^b=cg,f^b=fi,h^b=hi,d^c=de,f^d=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, 288446);