#125047

order	512
nilpotency class	4
isoclinism	isoclinic class of #124662
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	C₂×#102
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

372 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 102 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=fghi,d^2=gi,e^2=h,f^2=i,g^2=i,b^a=bd,c^a=cfgh,d^a=df,f^a=fi,g^a=gi,d^b=dg,e^b=efgi,f^b=fi,g^b=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 32243);