#505407

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

499 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)(17,18,23,28)(19,24,32,41)(20,25,22,27)(21,26,36,45)(29,38,31,40)(30,39,37,46)(33,42,35,44)(34,43,47,48);(2,5)(6,7)(9,11,12,16)(10,13,14,15)(17,19,23,32)(18,24,28,41)(20,30,22,37)(21,34,36,47)(25,39,27,46)(26,43,45,48)(29,35,31,33)(38,44,40,42);(10,14)(13,15)(17,20)(18,25)(19,29)(22,23)(24,38)(27,28)(30,34)(31,32)(33,35)(37,47)(39,43)(40,41)(42,44)(46,48);(1,3,4,8)(2,6,5,7)(10,14)(13,15)(17,20,23,22)(18,25,28,27)(19,30,32,37)(21,33,36,35)(24,39,41,46)(26,42,45,44)(29,47,31,34)(38,48,40,43)

Polycyclic presentation:

2^9|a^2=ei,b^2=fi,d^2=efi,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, 412603);