#418259

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

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

1267 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 44 points:

(1,2)(3,4)(5,6,9,8)(7,11,12,10)(13,14,18,24)(15,27,30,21)(16,32,17,34)(19,26,20,25)(22,36,23,33)(28,41,44,39)(29,43,35,38)(31,40,37,42);(5,7,9,12)(6,10,8,11)(13,15,18,30)(14,21,24,27)(16,29,17,35)(19,37,20,31)(22,39,23,41)(25,42,26,40)(28,36,44,33)(32,38,34,43);(2,4)(5,8,9,6)(7,11,12,10)(13,16,18,17)(14,22,24,23)(15,28,30,44)(19,36,20,33)(21,38,27,43)(25,34,26,32)(29,31,35,37)(39,40,41,42);(7,12)(10,11)(13,17,18,16)(14,23,24,22)(15,29,30,35)(19,36,20,33)(21,39,27,41)(25,34,26,32)(28,31,44,37)(38,40,43,42)

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=e,d^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,f^b=fi,g^b=gi,f^c=fi,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, 325455);