#474761

order	512
nilpotency class	3
isoclinism	isoclinic class of #467073
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2640
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#17006
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

664 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:

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

Polycyclic presentation:

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

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, 381957);