#450635

order	512
nilpotency class	3
isoclinism	isoclinic class of #417617
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2678
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15973
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

817 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:

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

Polycyclic presentation:

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

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, 357831);