#505398

order	512
nilpotency class	4
isoclinism	isoclinic class of #499724
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×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×D₈	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄ ◁ C₈×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

503 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2,4,5)(3,7,8,6)(9,10,12,14)(11,15,16,13)(17,18,21,27)(19,29,32,24)(20,25,23,26)(22,28,35,45)(30,41,33,39)(31,40,38,42)(34,43,37,44)(36,48,47,46);(2,5)(6,7)(9,11,12,16)(10,13,14,15)(17,19,21,32)(18,24,27,29)(20,31,23,38)(22,36,35,47)(25,42,26,40)(28,46,45,48)(30,37,33,34)(39,44,41,43);(10,14)(13,15)(17,20)(18,25)(19,30)(21,23)(24,39)(26,27)(29,41)(31,36)(32,33)(34,37)(38,47)(40,48)(42,46)(43,44);(1,3,4,8)(2,6,5,7)(10,14)(13,15)(17,20,21,23)(18,26,27,25)(19,31,32,38)(22,34,35,37)(24,40,29,42)(28,44,45,43)(30,47,33,36)(39,46,41,48)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 412594);