#508773

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	C₂×#466	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

583 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)(19,27)(20,21)(22,25)(23,26)(24,29)(28,36)(30,46)(31,44)(32,34)(33,35)(37,48)(38,43)(39,41)(40,47)(42,45);(2,5)(6,8)(10,13)(14,15)(17,19,21,29)(18,24,20,27)(22,36,34,45)(23,39,35,31)(25,42,32,28)(26,44,33,41)(30,48,47,38)(37,46,43,40);(3,7)(6,8)(9,11)(10,14)(12,16)(13,15)(17,20,21,18)(19,28,29,42)(22,37,34,43)(23,26,35,33)(24,36,27,45)(25,38,32,48)(30,44,47,41)(31,46,39,40);(1,3,4,7)(2,6,5,8)(11,16)(14,15)(18,20)(24,27)(25,32)(26,33)(28,42)(37,43)(40,46)(41,44)

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=ef,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, 415969);