#100795

order	512
nilpotency class	3
isoclinism	isoclinic class of #100715
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

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

353 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 353 conjugacy classes of subgroups — too many to draw. Only its 88 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) = 512

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=g,d^2=gh,e^2=hi,f^2=i,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, 7991);