#506529

order	512
nilpotency class	4
isoclinism	isoclinic class of #500839
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

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

774 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=eg,c^2=i,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=di,f^c=fh,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, 413725);