#508051

order	512
nilpotency class	4
isoclinism	isoclinic class of #508008
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1101 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)(7,10)(8,9)(11,12)(13,14)(15,26)(16,28)(17,22)(18,27)(19,39)(20,42)(21,32)(23,38)(24,41)(25,35)(29,40)(30,37)(31,34)(33,36)(43,44);(6,9)(10,12)(13,15,17,28)(14,16,22,26)(18,36,19,21)(20,43,35,31)(23,37,24,40)(25,44,42,34)(27,32,39,33)(29,41,30,38);(2,4)(5,7)(6,10)(8,11)(9,12)(13,16,17,26)(14,21,22,36)(15,27,28,39)(18,37,19,40)(20,44,35,34)(23,43,24,31)(25,30,42,29)(32,41,33,38);(7,11)(10,12)(14,22)(16,26)(23,24)(25,42)(27,39)(32,33)(34,44)(37,40)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 415247);