#100813

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₄³	C₂³
prime radical	C₂×#127	C₂×C₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

436 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 436 conjugacy classes of subgroups — too many to draw. Only its 80 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)(3,4)(7,8)(9,10,14,16,13,11,12,15)(17,18)(19,23,28,39)(20,30)(21,34)(22,25)(24,32)(26,36)(27,43,31,48)(29,46,45,38)(33,42)(35,41)(37,47,40,44);(1,2,3,8,5,7,4,6)(9,11,13,10)(12,15,14,16)(17,19)(18,23,25,39)(20,31)(21,29)(22,28)(24,40,41,37)(26,38,34,46)(27,33)(30,48,42,43)(32,47)(35,44)(36,45);(1,3,5,4)(2,6,7,8)(12,14)(15,16)(17,20,22,33)(18,24,25,41)(19,27,28,31)(21,35,36,32)(23,37,39,40)(26,42,34,30)(29,47,45,44)(38,48,46,43)

Polycyclic presentation:

2^9|a^2=f,b^2=dfgh,c^2=g,d^2=gh,e^2=hi,f^2=i,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, 8009);