#424597

order	512
nilpotency class	3
isoclinism	isoclinic class of #418470
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2676
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15323
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1189 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)(3,14,18,41)(4,10)(5,24)(6,12)(7,30)(8,13)(9,17,33,19)(11,28)(15,45,20,43)(16,32,48,58)(21,52)(22,36)(23,37)(25,39,49,44)(26,40)(27,42)(29,38)(31,47,34,50)(35,54)(46,59,51,57)(53,63,64,60)(55,61)(56,62);(1,3)(2,9)(4,20)(5,25)(6,19)(7,17)(8,18)(10,34)(11,39)(12,41)(13,33)(14,30)(15,23)(16,26)(21,53)(22,47)(24,32)(27,49)(28,48)(29,50)(31,37)(35,60)(36,45)(38,43)(40,44)(42,58)(46,56)(51,54)(52,57)(55,64)(59,61)(62,63);(1,4)(2,10)(3,15,18,20)(5,21)(6,29)(7,22)(8,23)(9,31,33,34)(11,35)(12,38)(13,37)(14,43,41,45)(16,46,48,51)(17,50,19,47)(24,52)(25,53,49,64)(26,56)(27,55)(28,54)(30,36)(32,57,58,59)(39,60,44,63)(40,62)(42,61);(1,5)(2,11)(3,16)(4,21)(6,28)(7,26)(8,27)(9,32)(10,35)(12,42)(13,40)(14,44)(15,46)(17,49)(18,48)(19,25)(20,51)(22,56)(23,55)(24,30)(29,54)(31,57)(33,58)(34,59)(36,52)(37,62)(38,61)(39,41)(43,63)(45,60)(47,53)(50,64)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 331793);