#437196

order	512
nilpotency class	3
isoclinism	isoclinic class of #420065
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2686
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15586
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

795 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:

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

Polycyclic presentation:

2^9|a^2=fg,c^2=ef,d^2=f,e^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,f^b=fi,g^b=gi,h^b=hi,g^c=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, 344392);