#508774

order	512
nilpotency class	4
isoclinism	isoclinic class of #508008
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	C₂⁴	C₂×D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

562 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)(3,6)(4,5)(7,8)(9,10)(11,15)(12,13)(14,16)(17,18)(19,22)(20,28)(21,29)(23,27)(24,26)(25,33)(30,37)(31,34)(32,41)(35,43)(36,38)(39,47)(40,44)(42,46)(45,48);(2,5)(6,8)(10,13)(14,15)(17,19,22,18)(20,31,33,42)(21,29,24,26)(23,38,37,44)(25,34,28,46)(27,40,30,36)(32,47,35,45)(39,41,48,43);(3,7)(6,8)(9,11)(10,14)(12,16)(13,15)(17,20,22,33)(18,25,19,28)(21,35,24,32)(23,39,37,48)(26,43,29,41)(27,45,30,47)(31,44,42,38)(34,36,46,40);(1,3,4,7)(2,6,5,8)(11,16)(14,15)(17,21,22,24)(18,26,19,29)(20,32,33,35)(23,36,37,40)(25,41,28,43)(27,44,30,38)(31,47,42,45)(34,48,46,39)

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=efi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,g^b=gh,h^b=hi,d^c=de,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, 415970);