#144215

order	512
nilpotency class	4
isoclinism	isoclinic class of #144213
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 ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

570 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 570 conjugacy classes of subgroups — too many to draw. Only its 66 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)(11,23,28,53)(12,29,33,46)(13,34,38,49)(14,21,15,20)(17,40,43,25)(18,39,47,32)(19,36,50,37)(24,54,57,71)(26,59,27,48)(30,62,56,52)(31,64,67,45)(35,51,58,42)(41,63,60,72)(44,65,69,55)(61,68,70,66);(1,3)(2,5)(4,7)(6,8)(9,11,16,28)(10,17,22,43)(12,30,33,56)(13,35,38,58)(14,26,15,27)(18,44,47,69)(19,48,50,59)(20,42,21,51)(23,49,53,34)(24,39,57,32)(25,36,40,37)(29,60,46,41)(31,65,67,55)(45,52,64,62)(54,70,71,61)(63,68,72,66);(5,8)(9,12)(10,18)(11,24)(13,31)(14,39)(15,32)(16,33)(17,41)(19,45)(20,46)(21,29)(22,47)(23,52)(25,55)(26,30)(27,56)(28,57)(34,61)(35,63)(36,68)(37,66)(38,67)(40,65)(42,69)(43,60)(44,51)(48,71)(49,70)(50,64)(53,62)(54,59)(58,72)

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=fh,g^a=gi,c^b=cf,d^b=dg,e^b=eh,f^b=fi,g^b=gi,e^c=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, 51411);