#475940

order	512
nilpotency class	3
isoclinism	isoclinic class of #457092
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

744 conjugacy classes of subgroups; only the 210 normal subgroups are shown.

normal subgroups only

This group has 744 conjugacy classes of subgroups — too many to draw. Only its 210 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) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=g,c^2=f,d^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^c=de,g^c=gi

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, 383136);