#463396

order	512
nilpotency class	3
isoclinism	isoclinic class of #456381
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2712
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16470
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

756 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:

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

Polycyclic presentation:

2^9|a^2=eg,c^2=ef,d^2=ef,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,h^b=hi,d^c=de,e^c=ei,f^c=fi,h^c=hi,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, 370592);