#506531

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	C₂×#2184	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

867 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)(13,14,17,22,30,38,44,36,16,21,29,37,18,23,31,20)(15,24,27,40,43,33,42,35,26,19,32,34,28,39,25,41);(5,7,8,12)(6,9,10,11)(13,15,17,27,30,43,44,42,16,26,29,32,18,28,31,25)(14,19,22,34,38,39,36,41,21,24,37,40,23,33,20,35);(2,4)(14,20)(15,25)(17,31)(18,30)(19,33)(21,36)(22,23)(24,39)(26,42)(27,28)(29,44)(32,43)(34,40)(37,38);(6,10)(9,11)(14,21)(15,26)(20,36)(22,37)(23,38)(25,42)(27,32)(28,43)

Polycyclic presentation:

2^9|a^2=f,b^2=egi,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, 413727);