#449113

order	512
nilpotency class	3
isoclinism	isoclinic class of #421157
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2610
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15931
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

718 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=e,d^2=eg,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,e^b=ei,h^b=hi,f^c=fi,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, 356309);