#456034

order	512
nilpotency class	3
isoclinism	isoclinic class of #453994
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2670
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16273
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

904 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=ef,c^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,h^b=hi,d^c=de,h^c=hi,e^d=ei,f^d=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, 363230);