#419772

order	512
nilpotency class	3
isoclinism	isoclinic class of #417692
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×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1577 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)(9,10,15,20,14,19,16,21)(11,22,24,33,18,13,31,32)(12,25,29,36,28,37,30,17)(23,39,26,27,35,34,40,38);(5,6)(7,8)(9,11)(10,13)(12,26)(14,18)(15,24)(16,31)(17,34)(19,22)(20,32)(21,33)(23,30)(25,38)(27,37)(28,40)(29,35)(36,39);(2,4)(9,12,14,28)(10,17,19,36)(11,23,18,35)(13,27,22,38)(15,30,16,29)(20,37,21,25)(24,40,31,26)(32,39,33,34);(5,7,8,6)(9,13)(10,18)(11,19)(12,27)(14,22)(15,33)(16,32)(17,35)(20,24)(21,31)(23,36)(25,26)(28,38)(29,39)(30,34)(37,40)

Polycyclic presentation:

2^9|a^2=f,c^2=i,d^2=ei,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,g^b=gi,f^c=fi,g^c=gi,h^c=hi,f^d=fi,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, 326968);