#463530

order	512
nilpotency class	3
isoclinism	isoclinic class of #455830
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2648
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16476
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

889 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:

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

Polycyclic presentation:

2^9|a^2=efg,c^2=ef,d^2=f,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,e^b=ei,d^c=de,e^c=ei,f^c=fi,g^c=gi,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, 370726);