#503048

order	512
nilpotency class	4
isoclinism	isoclinic class of #501064
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×#3176	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

817 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 410244);