#439345

order	512
nilpotency class	3
isoclinism	isoclinic class of #419458
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2709
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15635
prime radical	#23414	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1139 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:

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

Polycyclic presentation:

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

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, 346541);