#145043

order	512
nilpotency class	4
isoclinism	isoclinic class of #145027
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2034
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#449
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

307 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 307 conjugacy classes of subgroups — too many to draw. Only its 74 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) = 72
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 72 points:

(3,8)(5,7)(9,10,15,20)(11,21,23,16)(12,26,28,61)(13,30,34,64)(14,35,32,19)(17,40,41,27)(18,43,45,33)(22,50,51,60)(24,56,54,63)(25,39,55,49)(29,62,36,44)(31,46,65,42)(37,52,47,69)(38,58,48,70)(53,67,59,71)(57,66,72,68);(1,2)(3,5)(4,6)(7,8)(9,11,14,25,15,23,32,55)(10,16,19,39,20,21,35,49)(12,22,29,53,28,51,36,59)(13,24,33,58,34,54,43,70)(17,37,42,66,41,47,46,68)(18,38,30,63,45,48,64,56)(26,60,44,67,61,50,62,71)(27,52,31,57,40,69,65,72);(2,5)(3,8)(6,7)(9,12)(10,17)(11,22,23,51)(13,31)(14,36)(15,28)(16,37,21,47)(18,44)(19,46)(20,41)(24,57,54,72)(25,59,55,53)(26,30)(27,33)(29,32)(34,65)(35,42)(38,67,48,71)(39,68,49,66)(40,43)(45,62)(50,63,60,56)(52,70,69,58)(61,64)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=d,e^2=gi,f^2=h,g^2=i,b^a=bd,c^a=ce,e^a=eg,f^a=fh,g^a=gi,c^b=cf,d^b=di,e^b=eh,f^b=fhi,e^c=eg,f^c=fhi,g^c=gi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 52239);