#141741

order	512
nilpotency class	4
isoclinism	isoclinic class of #141698
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1972
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#496
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

279 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 279 conjugacy classes of subgroups — too many to draw. Only its 98 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfh,c^2=egh,d^2=gi,e^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^c=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 48937);