#144397

order	512
nilpotency class	4
isoclinism	isoclinic class of #144396
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
prime radical	#23898	C₂
LCS	G ▷ C₂²×#41 ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 847 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)(14,43)(16,49)(17,35)(19,55)(21,39)(23,32)(24,37)(27,34)(28,40)(30,59)(31,51)(33,56)(36,58)(38,48)(41,45)(42,52)(44,53)(46,63)(47,60)(50,62)(54,57)(61,64);(1,3)(2,8)(4,14)(5,23)(6,16)(7,17)(9,30)(10,35)(11,38)(12,32)(13,22)(15,25)(18,53)(19,47)(20,45)(21,46)(24,51)(26,31)(27,50)(28,48)(29,52)(33,62)(34,61)(36,41)(37,42)(39,60)(40,49)(43,55)(44,57)(54,63)(56,59)(58,64);(1,4,7,21)(2,9,12,34)(3,14,17,46)(5,19,25,57)(6,27,29,20)(8,30,32,61)(10,36,22,33)(11,39,26,18)(13,41,35,62)(15,47,23,44)(16,50,52,45)(24,58,28,56)(31,53,38,60)(37,55,40,54)(42,63,49,43)(48,64,51,59)

Polycyclic presentation:

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

Conjugacy classes & character table

35 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285029
maximal normal subgroup of	#72285029

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 51593);