#144222

order	512
nilpotency class	4
isoclinism	isoclinic class of #144222
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#10320
commutator, Frattini, cosocle	C₂×#131	C₂³
LCS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#131 ◁ G
LECS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁

582 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 582 conjugacy classes of subgroups — too many to draw. Only its 62 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) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72341938, #72341945
maximal normal subgroup of	#72341945

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 51418);