#100877

order	512
nilpotency class	3
isoclinism	isoclinic class of #100827
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

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

347 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 347 conjugacy classes of subgroups — too many to draw. Only its 86 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) = 512

A minimal faithful permutation representation — 3 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfgi,c^2=h,d^2=gh,e^2=ghi,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=egh,d^b=dh,e^b=ei,f^b=fh,d^c=di,e^c=eghi,f^c=fi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8073);