#449601

order	512
nilpotency class	3
isoclinism	isoclinic class of #418871
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2697
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15948
prime radical	#60398	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1967 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=eh,h^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,f^b=fi,e^c=ei,h^c=hi,f^d=fi,g^d=gi,h^d=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 356797);