#317948

order	512
nilpotency class	3
isoclinism	isoclinic class of #317945
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2537
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3169	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

523 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 523 conjugacy classes of subgroups — too many to draw. Only its 138 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)(3,7)(4,8)(5,9)(6,10)(11,16)(12,15)(13,14)(17,18,22,28)(19,29,32,25)(20,33,35,39)(21,24,23,27)(26,36,42,34)(30,41,37,38)(31,45,46,47)(40,44,43,48);(1,3,5,12,6,13,4,11)(2,7,9,15,10,14,8,16)(18,24,28,27)(19,30,32,37)(25,29)(26,39,42,33)(31,44,46,48)(38,41)(40,43)(45,47);(2,8)(3,11)(4,5)(7,14)(9,10)(12,13)(17,19)(18,25)(20,31)(21,30)(22,32)(23,37)(24,38)(26,40)(27,41)(28,29)(33,47)(34,44)(35,46)(36,48)(39,45)(42,43);(17,20,23,36,22,35,21,34)(18,26,24,39,28,42,27,33)(19,31,37,48,32,46,30,44)(25,40,38,45,29,43,41,47)

Polycyclic presentation:

2^9|a^2=h,b^2=eg,d^2=eh,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=cg,f^c=fi,g^c=ghi

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, 225144);