#413899

order	512
nilpotency class	3
isoclinism	isoclinic class of #400688
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2544
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15159
prime radical	#23414	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1368 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)(3,8)(4,9)(5,22)(6,11)(7,13)(10,24)(12,27)(14,40)(15,43)(16,30)(17,31)(18,47)(19,49)(20,35)(21,36)(23,39)(25,32)(26,37)(28,42)(29,45)(33,46)(34,51)(38,54)(41,59)(44,52)(48,61)(50,53)(55,58)(56,60)(57,63)(62,64);(1,3)(2,8)(4,18)(5,15)(6,25)(7,16)(9,33)(10,29)(11,32)(12,31)(13,30)(14,21)(17,27)(19,48)(20,46)(22,52)(23,44)(24,55)(26,42)(28,36)(34,59)(35,47)(37,40)(38,58)(39,43)(41,51)(45,54)(49,61)(50,63)(53,57)(56,62)(60,64);(1,4)(2,9)(3,14)(5,19)(6,26)(7,20)(8,28)(10,34)(11,37)(12,36)(13,35)(15,41)(16,42)(17,46)(18,25)(21,27)(22,53)(23,50)(24,56)(29,57)(30,40)(31,47)(32,33)(38,60)(39,49)(43,59)(44,62)(45,63)(48,55)(51,54)(52,64)(58,61);(1,5,7,23)(2,10)(3,15,16,44)(4,19,20,50)(6,24,27,54)(8,29)(9,34)(11,39)(12,22)(13,38)(14,41,42,62)(17,45,25,55)(18,48,46,63)(21,51,26,56)(28,57)(30,58)(31,52)(32,43)(33,59)(35,60)(36,53)(37,49)(40,61)(47,64)

Polycyclic presentation:

2^9|d^2=gh,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,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

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, 321095);