#505142

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

531 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:

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

Polycyclic presentation:

2^9|a^2=ei,b^2=efi,c^2=i,d^2=f,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, 412338);