#427263

order	512
nilpotency class	3
isoclinism	isoclinic class of #427262
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15368
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1308 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319923, #72319988, #72320088
maximal normal subgroup of	#72319923, #72319988, #72320088

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 334459);