#418295

order	512
nilpotency class	3
isoclinism	isoclinic class of #417603
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	#23390	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1708 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)(11,21)(12,17)(13,19)(14,20)(15,31)(16,24)(18,28)(22,33)(23,36)(25,35)(26,40)(27,37)(29,34)(30,38)(32,39);(5,6)(7,8)(9,11)(10,16)(12,26)(13,24)(14,29)(15,25)(17,36)(18,32)(19,21)(20,35)(22,28)(23,27)(30,39)(31,34)(33,38)(37,40);(2,4)(9,12)(10,17)(11,22)(13,27)(14,30)(15,28)(16,32)(18,31)(19,37)(20,38)(21,33)(23,25)(24,39)(26,29)(34,40)(35,36);(5,7,8,6)(9,12)(10,18)(11,23)(13,27)(14,28)(15,30)(16,33)(17,31)(19,38)(20,37)(21,32)(22,25)(24,26)(29,39)(34,40)(35,36)

Polycyclic presentation:

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

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, 325491);