#395465

order	512
nilpotency class	3
isoclinism	isoclinic class of #394482
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2435
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14724
prime radical	#63090	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3384 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)(3,14,17,9)(4,20,23,10)(5,26)(6,13)(7,29)(8,12)(11,28)(15,41)(16,46,47,58)(18,39,19,33)(21,50)(22,53,54,60)(24,40,25,36)(27,37)(30,38)(31,44)(32,48,42,49)(34,45)(35,55,51,56)(43,59)(52,61)(57,63)(62,64);(1,3)(2,9)(4,21)(5,16)(6,19)(7,18)(8,17)(10,34)(11,32)(12,14)(13,39)(15,24)(20,50)(22,52)(23,45)(25,44)(26,46)(27,47)(28,49)(29,33)(30,48)(31,36)(35,59)(37,58)(38,42)(40,41)(43,55)(51,64)(53,57)(54,63)(56,62)(60,61);(1,4)(2,10)(3,15)(5,22)(6,25)(7,24)(8,23)(9,31)(11,35)(12,20)(13,40)(14,41)(16,43)(17,44)(18,45)(19,21)(26,53)(27,54)(28,56)(29,36)(30,55)(32,57)(33,50)(34,39)(37,60)(38,51)(42,61)(46,64)(47,62)(48,63)(49,52)(58,59);(1,5)(2,11)(3,16)(4,22)(6,28)(7,30)(8,27)(9,32)(10,35)(12,38)(13,37)(14,42)(15,43)(17,47)(18,48)(19,49)(20,51)(21,52)(23,54)(24,55)(25,56)(26,29)(31,57)(33,46)(34,59)(36,53)(39,58)(40,60)(41,61)(44,62)(45,63)(50,64)

Polycyclic presentation:

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

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, 302661);