#137229

order	512
nilpotency class	4
isoclinism	isoclinic class of #12108
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#447
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#408 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

455 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 455 conjugacy classes of subgroups — too many to draw. Only its 94 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) = 52
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 52 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfgh,d^2=gi,e^2=i,f^2=gi,g^2=i,b^a=bd,c^a=ce,e^a=ehi,d^b=dg,e^b=eh,f^b=fg,g^b=gi,d^c=dh,e^c=ei,f^c=fh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 44425);