#395228

order	512
nilpotency class	3
isoclinism	isoclinic class of #393925
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1775 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 302424);