#430895

order	512
nilpotency class	3
isoclinism	isoclinic class of #418438
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1113 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=fi,d^2=ef,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=de,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, 338091);