#144223

order	512
nilpotency class	4
isoclinism	isoclinic class of #144222
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#10320
commutator, Frattini, cosocle	C₂×#131	C₂³
prime radical	#32948	C₂
LCS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#131 ◁ G
LECS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 568 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:

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

Polycyclic presentation:

2^9|a^2=i,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	#72341941, #72341945, #72341992
maximal normal subgroup of	#72341945

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 51419);