#508915

order	512
nilpotency class	4
isoclinism	isoclinic class of #508304
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×#466	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

729 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)(7,10)(8,9)(11,12)(13,14,19,30,29,39,28,21,17,22,16,23,18,24,32,38)(15,20,31,43,40,44,42,37,25,34,26,35,27,36,41,33);(1,2,3,4)(6,9)(10,12)(13,15)(14,20)(16,26)(17,25)(18,27)(19,31)(21,37)(22,34)(23,35)(24,36)(28,42)(29,40)(30,43)(32,41)(33,38)(39,44);(2,4)(5,7)(6,10)(8,11)(9,12)(14,21)(15,25)(16,28)(18,29)(19,32)(20,33)(22,38)(23,24)(26,41)(30,39)(31,42)(34,37)(35,44)(36,43);(7,11)(10,12)(14,22)(20,34)(21,38)(23,30)(24,39)(33,37)(35,43)(36,44)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 416111);