#471022

order	512
nilpotency class	3
isoclinism	isoclinic class of #462392
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2517
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16734
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

510 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:

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

Polycyclic presentation:

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

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, 378218);