#435742

order	512
nilpotency class	3
isoclinism	isoclinic class of #420065
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2686
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15536
prime radical	C₂⁴	C₂²×Q₈
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

591 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=ef,c^2=ef,d^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^b=hi,g^c=gi,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, 342938);