#280859

order	512
nilpotency class	3
isoclinism	isoclinic class of #280843
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2420
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂³	#546
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

430 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 126 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=i,d^2=eh,e^2=h,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fh,c^b=cg,g^b=gi,f^c=fh,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, 188055);