#422912

order	512
nilpotency class	3
isoclinism	isoclinic class of #417603
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3045	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1940 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:

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

Polycyclic presentation:

2^9|b^2=e,c^2=f,d^2=i,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=de,g^b=gi,g^c=gi,e^d=ei,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, 330108);