#100804

order	512
nilpotency class	3
isoclinism	isoclinic class of #100723
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

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

normal subgroups only

This group has 335 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) = 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)(27,31,36,47)(28,38)(29,42)(30,33)(32,40)(34,44)(35,51,39,56)(37,54,53,46)(41,50)(43,49)(45,55,48,52);(1,2,4,6,5,7,3,8)(9,11)(10,15)(12,22)(13,20)(14,21)(16,23)(17,24)(18,19)(25,27)(26,31,33,47)(28,39)(29,37)(30,36)(32,48,49,45)(34,46,42,54)(35,41)(38,56,50,51)(40,55)(43,52)(44,53);(1,3,5,4)(2,6,7,8)(10,16)(12,13)(15,23)(20,22)(25,28,30,41)(26,32,33,49)(27,35,36,39)(29,43,44,40)(31,45,47,48)(34,50,42,38)(37,55,53,52)(46,56,54,51)

Polycyclic presentation:

2^9|a^2=f,b^2=df,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, 8000);