#142917

order	512
nilpotency class	4
isoclinism	isoclinic class of #142236
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#447
commutator, Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ #1729 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

433 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 433 conjugacy classes of subgroups — too many to draw. Only its 80 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=i,d^2=gi,f^2=gi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,f^a=fhi,g^a=gi,h^a=hi,c^b=cf,d^b=dg,f^b=fg,g^b=gi,h^b=hi,d^c=dh,f^c=fg,g^c=gi,h^c=hi

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, 50113);