#100818

order	512
nilpotency class	3
isoclinism	isoclinic class of #100715
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#123
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

395 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 395 conjugacy classes of subgroups — too many to draw. Only its 76 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) = 56
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 56 points:

(1,2,4,7,5,8,3,6)(9,10)(11,14,18,22)(12,19,20,15)(13,16)(17,21)(23,24)(25,26)(27,36,39,47)(28,40)(29,34)(30,44)(31,35)(32,38,48,43)(33,41)(37,52,55,46)(42,49)(45,50)(51,53,56,54);(2,6,8,7)(9,11,13,18)(10,14)(12,17,20,23)(15,21)(16,22)(19,24)(25,27,30,38,31,39,29,43)(26,32,44,36,35,48,34,47)(28,37,45,53,42,55,41,54)(33,46,40,51,50,52,49,56);(9,12,13,20)(10,15,16,19)(11,17,18,23)(14,21,22,24)(25,28,31,42)(26,33,35,50)(27,37,39,55)(29,41,30,45)(32,46,48,52)(34,49,44,40)(36,51,47,56)(38,53,43,54)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8014);