#133840

order	512
nilpotency class	4
isoclinism	isoclinic class of #133839
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#8541
commutator	C₂²×D₄	C₂²×C₄
Frattini, cosocle	C₂²×#41	C₂³
prime radical	#60401	C₂
LCS	G ▷ C₂²×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂³ ▷ C₂ ▷ C₁

1401 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 1401 conjugacy classes of subgroups — too many to draw. Only its 88 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72209847, #72342796
maximal normal subgroup of	#72209847, #72342796
maximal quotient of	#72227684, #72227722, #72227763, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 41036);