#120649

order	512
nilpotency class	3
isoclinism	isoclinic class of #10557
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#394
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1442
prime radical	C₂⁴	C₄×Q₈
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 392 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 — 3 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eh,c^2=d,d^2=hi,e^2=hi,f^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,c^b=cf,d^b=di,f^b=fh,g^b=gi,e^c=ei,f^c=fh,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, 27845);