#6437592

order	512
nilpotency class	3
isoclinism	isoclinic class of #23892
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹¹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2403
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂	C₂×#2403
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

5892 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)(3,15)(4,21)(5,11)(6,12)(7,13)(8,14)(9,19)(10,25)(16,45)(17,44)(18,34)(20,33)(22,43)(23,51)(24,38)(26,37)(27,32)(28,40)(29,42)(30,41)(31,48)(35,47)(36,54)(39,50)(46,62)(49,57)(52,61)(53,60)(55,59)(56,64)(58,63);(1,3)(2,9)(4,22)(5,27)(6,18)(7,19)(8,20)(10,35)(11,39)(12,15)(13,33)(14,34)(16,24)(17,28)(21,31)(23,46)(25,48)(26,47)(29,49)(30,50)(32,40)(36,56)(37,45)(38,43)(41,57)(42,44)(51,61)(52,54)(53,63)(55,64)(58,59)(60,62);(1,4)(2,10)(3,16)(5,23)(6,24)(7,25)(8,26)(9,31)(11,36)(12,21)(13,37)(14,38)(15,43)(17,46)(18,47)(19,22)(20,48)(27,52)(28,53)(29,55)(30,54)(32,56)(33,35)(34,45)(39,58)(40,51)(41,59)(42,60)(44,61)(49,63)(50,64)(57,62);(1,5)(2,11)(3,17)(4,23)(6,28)(7,30)(8,29)(9,32)(10,36)(12,40)(13,41)(14,42)(15,44)(16,46)(18,49)(19,27)(20,50)(21,51)(22,52)(24,53)(25,54)(26,55)(31,56)(33,39)(34,57)(35,58)(37,59)(38,60)(43,61)(45,62)(47,63)(48,64);(1,6,8,7)(2,12,14,13)(3,18,20,19)(4,24,26,25)(5,28,29,30)(9,15,34,33)(10,21,38,37)(11,40,42,41)(16,47,48,22)(17,49,50,27)(23,53,55,54)(31,43,45,35)(32,44,57,39)(36,51,60,59)(46,63,64,52)(56,61,62,58)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72312363, #72312375, #72312530
maximal normal subgroup of	#72312363, #72312375, #72312530

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6344788);