#423171

order	512
nilpotency class	3
isoclinism	isoclinic class of #420065
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2686
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15312
prime radical	#60469	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1278 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)(4,10)(5,25)(6,14)(7,13)(8,12)(9,18)(11,29)(16,42)(17,32)(19,41)(20,49)(21,51)(22,36)(23,37)(24,35)(26,43)(27,40)(28,39)(30,33)(31,47)(34,55)(38,48)(44,63)(45,59)(46,56)(50,58)(52,62)(53,61)(54,57)(60,64);(1,3)(2,9)(4,20)(5,26)(6,30)(7,19)(8,18)(10,31)(11,38)(12,41)(13,33)(14,15)(16,23)(17,27)(21,52)(22,45)(24,47)(25,56)(28,48)(29,46)(32,39)(34,60)(35,42)(36,49)(37,59)(40,43)(44,55)(50,53)(51,63)(54,64)(57,58)(61,62);(1,4,8,22,7,23,6,24)(2,10,14,35,13,37,12,36)(3,16,30,45,19,20,18,47)(5,21,27,55,28,54,29,53)(9,31,41,49,33,59,15,42)(11,34,39,51,40,61,25,57)(17,44,26,52,46,50,48,64)(32,58,38,62,56,63,43,60);(1,5)(2,11)(3,17)(4,21)(6,29)(7,28)(8,27)(9,32)(10,34)(12,25)(13,40)(14,39)(15,43)(16,44)(18,48)(19,46)(20,50)(22,55)(23,54)(24,53)(26,30)(31,58)(33,56)(35,51)(36,57)(37,61)(38,41)(42,60)(45,52)(47,64)(49,62)(59,63)

Polycyclic presentation:

2^9|c^2=ef,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,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, 330367);