#6434131

order	512
nilpotency class	3
isoclinism	isoclinic class of #6434130
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#2631
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#23412	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1903 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=i,d^2=fg,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,h^a=hi,c^b=ch,e^b=ei,f^b=fi,g^b=gi,h^b=hi,h^c=hi,f^d=fi,g^d=gi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72264914, #72264954, #72265007, …
maximal normal subgroup of	#72264914, #72264954, #72265007, …
maximal quotient of	#72265528

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6341327);