#450593

order	512
nilpotency class	3
isoclinism	isoclinic class of #417617
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2678
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15972
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

817 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)(3,15,19,40)(4,20)(5,25,28,41)(6,12)(7,13)(8,14)(9,18,33,29)(10,23)(11,27,39,30)(16,42,45,60)(17,32)(21,49,52,55)(22,37)(24,36)(26,38)(31,44,34,53)(35,54,48,51)(43,56)(46,58)(47,57)(50,62)(59,64)(61,63);(1,3,8,18,7,19,6,29)(2,9,14,15,13,33,12,40)(4,16,24,44,23,45,22,53)(5,26,30,47,28,46,27,17)(10,34,37,60,20,31,36,42)(11,38,41,57,39,58,25,32)(21,50,54,64,52,63,51,43)(35,61,49,56,48,62,55,59);(1,4,8,24,7,23,6,22)(2,10,14,37,13,20,12,36)(3,16,18,44,19,45,29,53)(5,21,30,54,28,52,27,51)(9,31,15,42,33,34,40,60)(11,35,41,49,39,48,25,55)(17,43,26,50,47,64,46,63)(32,56,38,62,57,59,58,61);(1,5,6,27,7,28,8,30)(2,11,12,25,13,39,14,41)(3,17,18,26,19,47,29,46)(4,21,22,51,23,52,24,54)(9,32,15,38,33,57,40,58)(10,35,36,55,20,48,37,49)(16,43,44,50,45,64,53,63)(31,56,42,62,34,59,60,61)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 357789);