#144362

order	512
nilpotency class	4
isoclinism	isoclinic class of #144362
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#10336
commutator, Frattini, cosocle	C₂²×#41	C₂³
LCS	G ▷ C₂²×#41 ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂³ ▷ C₂ ▷ C₁

872 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 872 conjugacy classes of subgroups — too many to draw. Only its 54 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)(3,13)(4,18)(5,10)(6,11)(7,12)(8,15)(9,20)(14,41)(16,39)(17,40)(19,51)(21,52)(22,33)(23,34)(24,36)(25,37)(26,58)(27,46)(28,47)(29,61)(30,56)(31,57)(32,53)(35,48)(38,43)(42,50)(44,64)(45,59)(49,60)(54,63)(55,62);(1,3)(2,8)(4,14)(5,15)(6,16)(7,17)(9,29)(10,13)(11,35)(12,28)(18,50)(19,42)(20,55)(21,44)(22,46)(23,47)(24,48)(25,49)(26,31)(27,37)(30,62)(32,38)(33,60)(34,40)(36,39)(41,51)(43,57)(45,54)(52,59)(53,58)(56,61)(63,64);(1,4)(2,9)(3,14)(5,19)(6,20)(7,21)(8,26)(10,32)(11,18)(12,31)(13,38)(15,45)(16,43)(17,44)(22,53)(23,54)(24,56)(25,57)(27,59)(28,29)(30,34)(33,51)(35,50)(36,63)(37,52)(39,41)(40,62)(42,47)(46,58)(48,61)(49,55)(60,64)

Polycyclic presentation:

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

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72209847, #72258340
maximal normal subgroup of	#72209847, #72258340
maximal quotient of	#72227678, #72227721, #72227934, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 51558);