#412074

order	512
nilpotency class	3
isoclinism	isoclinic class of #396100
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2414
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15126
prime radical	C₂×#3067	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1299 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)(3,15)(4,20)(5,26)(6,13)(7,12)(8,14)(9,18)(10,24)(11,28)(16,35)(17,45)(19,33)(21,49)(22,52)(23,37)(25,38)(27,40)(29,39)(30,34)(31,51)(32,47)(36,53)(41,44)(42,46)(43,58)(48,55)(50,60)(54,56)(57,59)(61,64)(62,63);(1,3)(2,9)(4,21)(5,17)(6,30)(7,18)(8,19)(10,35)(11,32)(12,33)(13,15)(14,34)(16,23)(20,41)(22,50)(24,51)(25,44)(26,42)(27,57)(28,46)(29,47)(31,38)(36,60)(37,49)(39,59)(40,45)(43,53)(48,61)(52,63)(54,64)(55,62)(56,58);(1,4,6,23,8,24,7,25)(2,10,12,37,14,20,13,38)(3,16,18,21,19,44,30,51)(5,22,27,53,29,54,28,55)(9,31,15,41,34,49,33,35)(11,36,39,52,40,48,26,56)(17,43,46,50,47,62,57,64)(32,58,42,61,45,63,59,60);(1,5,7,28,8,29,6,27)(2,11,13,26,14,40,12,39)(3,17,18,46,19,47,30,57)(4,22,25,55,24,54,23,53)(9,32,15,42,34,45,33,59)(10,36,38,56,20,48,37,52)(16,43,21,50,44,62,51,64)(31,58,41,61,49,63,35,60)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 319270);