#447639

order	512
nilpotency class	3
isoclinism	isoclinic class of #419454
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1050 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 354835);