#405413

order	512
nilpotency class	3
isoclinism	isoclinic class of #395586
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2411
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14912
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1218 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)(4,21)(5,11)(6,13)(7,12)(8,14)(9,18)(10,24)(16,29)(17,41)(19,32)(20,31)(22,51)(23,36)(25,35)(26,39)(27,38)(28,37)(30,46)(33,45)(34,52)(40,44)(42,58)(43,49)(47,57)(48,56)(50,53)(54,59)(55,62)(60,61)(63,64);(1,3)(2,9)(4,16,25,45)(5,17)(6,19)(7,18)(8,20)(10,33,36,29)(11,30)(12,31)(13,15)(14,32)(21,49,35,40)(22,42,53,62)(23,44,24,43)(26,48)(27,47)(28,46)(34,58,59,55)(37,57)(38,56)(39,41)(50,64,51,60)(52,63,54,61);(1,4,6,23,8,25,7,24)(2,10,12,35,14,36,13,21)(3,16,18,43,20,45,19,44)(5,22,26,52,27,53,28,54)(9,29,15,40,32,33,31,49)(11,34,37,51,38,59,39,50)(17,42,46,61,47,62,48,63)(30,55,41,60,56,58,57,64);(1,5)(2,11)(3,17)(4,22)(6,26)(7,28)(8,27)(9,30)(10,34)(12,37)(13,39)(14,38)(15,41)(16,42)(18,46)(19,48)(20,47)(21,50)(23,52)(24,54)(25,53)(29,55)(31,57)(32,56)(33,58)(35,51)(36,59)(40,60)(43,61)(44,63)(45,62)(49,64)

Polycyclic presentation:

2^9|b^2=h,c^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,g^c=gi,h^c=hi,f^d=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, 312609);