#512247

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	C₂²×D₁₆	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #376 ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

564 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 140 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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

Polycyclic presentation:

2^9|b^2=efi,c^2=ei,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, 419443);