#6371102

order	512
nilpotency class	3
isoclinism	isoclinic class of #20671
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	C₂×#117
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂	C₂×#2283
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1689 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)(3,13,16,35)(4,18,21,9)(5,10)(6,11)(7,12)(8,17,27,42)(14,36)(15,34,40,55)(19,44,47,28)(20,45,48,29)(22,30)(23,31)(24,32)(25,39)(26,41,51,60)(33,38)(37,56)(43,59)(46,62,63,52)(49,53)(50,58)(54,57)(61,64);(1,3)(2,8,12,27)(4,14)(5,15)(6,17)(7,16)(9,25,18,43)(10,26,31,51)(11,13,32,35)(19,37)(20,39)(21,38)(22,41)(23,40)(24,42)(28,50,44,61)(29,33,45,36)(30,34,53,55)(46,58)(47,57)(48,59)(49,60)(52,54,62,56)(63,64);(1,4)(2,9)(3,14)(5,19)(6,20)(7,21)(8,25)(10,28)(11,29)(12,18)(13,33)(15,37)(16,38)(17,39)(22,46)(23,47)(24,48)(26,50)(27,43)(30,52)(31,44)(32,45)(34,54)(35,36)(40,57)(41,58)(42,59)(49,63)(51,61)(53,62)(55,56)(60,64);(1,5,6,22,7,23,24,49)(2,10,11,30,12,31,32,53)(3,15,17,41,16,40,42,60)(4,19,20,46,21,47,48,63)(8,26,13,34,27,51,35,55)(9,28,29,52,18,44,45,62)(14,37,39,58,38,57,59,64)(25,50,33,54,43,61,36,56);(3,16)(8,27)(13,35)(14,38)(15,40)(17,42)(25,43)(26,51)(33,36)(34,55)(37,57)(39,59)(41,60)(50,61)(54,56)(58,64)

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285642, #72285922, #72286021, …
maximal normal subgroup of	#72285642, #72285922, #72286021, …
maximal quotient of	C₂×#6371102

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6278298);