#512347

order	512
nilpotency class	4
isoclinism	isoclinic class of #511747
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

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

529 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=ef,c^2=i,d^2=e,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dg,g^a=gh,h^a=hi,c^b=ci,g^d=gh,h^d=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419543);