#506504

order	512
nilpotency class	4
isoclinism	isoclinic class of #500829
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	C₂³×D₈	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

862 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)(7,11)(8,10)(9,12)(13,14,19,31,30,39,29,21,18,24,16,22,17,23,33,38)(15,20,32,42,41,43,25,37,28,36,26,35,27,34,40,44);(5,7,8,12)(6,9,10,11)(13,15,16,26,30,41,33,40,18,28,19,32,17,27,29,25)(14,20,22,35,39,43,38,44,24,36,31,42,23,34,21,37);(2,4)(14,21)(15,25)(16,29)(17,30)(19,33)(20,34)(22,23)(24,38)(26,27)(28,40)(31,39)(32,41)(35,42)(36,43);(6,10)(9,11)

Polycyclic presentation:

2^9|a^2=fi,b^2=eg,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,f^c=fh,g^c=gh,h^c=hi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 413700);