#137048

order	512
nilpotency class	4
isoclinism	isoclinic class of #12316
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#447
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1874
prime radical	C₂×#41	D₈
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₈×#41 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

316 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 90 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 — 3 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfgi,c^2=i,d^2=gi,e^2=h,f^2=ghi,g^2=i,b^a=bd,c^a=ce,d^b=dg,e^b=eh,f^b=fg,g^b=gi,d^c=dh,e^c=eh,f^c=fh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 44244);