#471191

order	512
nilpotency class	3
isoclinism	isoclinic class of #455830
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2648
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16774
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

655 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:

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

Polycyclic presentation:

2^9|a^2=eg,b^2=h,c^2=f,d^2=ef,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,e^b=ei,f^b=fi,d^c=de,e^c=ei,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, 378387);