#100823

order	512
nilpotency class	3
isoclinism	isoclinic class of #100723
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	C₂²×#41	D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

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

normal subgroups only

This group has 436 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) = 56
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfh,d^2=gh,e^2=hi,f^2=gi,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, 8019);