#430344

order	512
nilpotency class	3
isoclinism	isoclinic class of #418473
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2673
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15418
prime radical	C₂×#2017	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1141 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,d^2=ef,e^2=i,f^2=i,g^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,h^b=hi,e^c=ei,f^c=fi,g^c=gi,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, 337540);