#474701

order	512
nilpotency class	3
isoclinism	isoclinic class of #455690
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2825
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16995
prime radical	C₂⁴	C₂²×Q₈
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

551 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=egh,c^2=g,d^2=ef,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,d^c=de,f^c=fi,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, 381897);