#144365

order	512
nilpotency class	4
isoclinism	isoclinic class of #144365
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

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

normal subgroups only

This group has 856 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)(5,22)(6,26)(7,12)(8,15)(9,20)(10,25)(11,29)(14,44)(16,50)(17,36)(19,56)(21,40)(23,32)(24,38)(27,35)(28,41)(30,60)(31,52)(33,62)(34,57)(37,59)(39,49)(42,46)(43,53)(45,54)(47,64)(48,61)(51,63)(55,58);(1,3)(2,8)(4,14)(5,23)(6,16)(7,17)(9,33)(10,36)(11,39)(12,32)(13,22)(15,25)(18,54)(19,48)(20,51)(21,47)(24,52)(26,31)(27,46)(28,49)(29,53)(30,35)(34,42)(37,63)(38,43)(40,61)(41,50)(44,56)(45,58)(55,64)(57,62)(59,60);(1,4)(2,9)(3,14)(5,19)(6,27)(7,21)(8,30)(10,37)(11,40)(12,35)(13,42)(15,48)(16,51)(17,47)(18,26)(20,29)(22,34)(23,45)(24,59)(25,58)(28,57)(31,54)(32,33)(36,63)(38,56)(39,61)(41,55)(43,64)(44,50)(46,53)(49,62)(52,60)

Polycyclic presentation:

2^9|d^2=i,e^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,f^a=fghi,g^a=gi,h^a=hi,c^b=cf,d^b=di,e^b=eh,g^b=gi,d^c=dg,e^c=ei,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	#72317516
maximal normal subgroup of	#72317516

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 51561);