#474604

order	512
nilpotency class	3
isoclinism	isoclinic class of #453995
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2812
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16987
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

762 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 381800);