#506540

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	#25777	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

819 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)(13,14,20,27,17,37,38,43,18,25,19,26,21,24,33,44)(15,28,31,23,34,40,35,41,29,22,30,39,16,32,36,42);(5,7,8,12)(6,9,10,11)(13,15,20,31,17,34,38,35,18,29,19,30,21,16,33,36)(14,22,27,39,37,32,43,42,25,28,26,23,24,40,44,41);(2,4)(13,16)(14,23)(15,21)(17,29)(18,34)(19,31)(20,30)(22,26)(24,41)(25,39)(27,28)(32,43)(33,36)(35,38)(37,42)(40,44);(6,10)(9,11)(13,17,18,21)(14,24,25,37)(15,16,29,34)(19,33,20,38)(22,32,28,40)(23,41,39,42)(26,43,27,44)(30,35,31,36)

Polycyclic presentation:

2^9|a^2=fi,b^2=egi,d^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, 413736);