#100744

order	512
nilpotency class	3
isoclinism	isoclinic class of #100723
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	#386
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₄³	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

316 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 72 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) = 512

A minimal faithful permutation representation — 3 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfi,c^2=g,d^2=gh,e^2=hi,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eh,c^b=cg,d^b=dh,e^b=ei,f^b=fh,d^c=di,e^c=ehi,f^c=fi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7940);