#506496

order	512
nilpotency class	4
isoclinism	isoclinic class of #501064
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×#2186	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=egi,d^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=di,f^c=fh,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, 413692);