#499966

order	512
nilpotency class	4
isoclinism	isoclinic class of #499712
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#3339	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

969 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)(15,23)(16,19)(17,29)(18,32)(20,24)(21,30)(22,28)(25,36)(26,43)(27,38)(31,40)(33,39)(34,42)(35,41)(37,44);(5,7,8,12)(6,9,10,11)(14,19)(15,24)(17,30)(18,28)(20,36)(21,38)(23,34)(25,42)(26,44)(27,41)(29,35)(31,33)(37,39)(40,43);(2,4)(13,15,16,25)(14,20,19,34)(17,31,27,44)(18,33,28,26)(21,39,35,43)(22,40,32,37)(23,38,36,29)(24,41,42,30);(6,10)(9,11)(14,19)(20,34)(21,35)(22,32)(23,36)(29,38)(37,40)(39,43)

Polycyclic presentation:

2^9|b^2=ei,c^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, 407162);