#506438

order	512
nilpotency class	4
isoclinism	isoclinic class of #500839
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈ ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

526 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=eg,c^2=i,d^2=ei,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=di,g^c=gh,h^c=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 413634);