#508050

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	C₂×#3155	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1197 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 44 points:

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

Polycyclic presentation:

2^9|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, 415246);