#500119

order	512
nilpotency class	4
isoclinism	isoclinic class of #499712
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#3287	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1043 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)(14,20)(15,24)(16,19)(18,30)(21,37)(23,40)(25,35)(26,44)(27,42)(28,29)(31,36)(32,34)(38,43)(39,41);(6,9)(10,12)(13,14)(15,25)(16,22)(17,20)(18,31)(19,33)(21,24)(23,30)(26,43)(27,37)(28,36)(29,40)(32,38)(34,41)(35,42)(39,44);(2,4)(5,7,8,11)(6,10,9,12)(13,15,17,27)(14,21,20,35)(16,26,19,34)(18,32,29,44)(22,38,33,39)(23,41,36,43)(24,28,42,30)(25,40,37,31);(6,9)(10,12)(13,16,17,19)(14,22,20,33)(15,26,27,34)(18,28,29,30)(21,38,35,39)(23,31,36,40)(24,32,42,44)(25,43,37,41)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 407315);