#450226

order	512
nilpotency class	3
isoclinism	isoclinic class of #418851
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2679
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15961
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1051 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:

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

Polycyclic presentation:

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

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, 357422);