#142255

order	512
nilpotency class	4
isoclinism	isoclinic class of #142252
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1999
commutator, Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#408 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

710 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 710 conjugacy classes of subgroups — too many to draw. Only its 72 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=gi,e^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,f^a=fhi,g^a=gi,c^b=cf,d^b=dg,e^b=ei,f^b=fi,g^b=gi,d^c=dh,e^c=ei,f^c=fi

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, 49451);