#450424

order	512
nilpotency class	3
isoclinism	isoclinic class of #417617
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2678
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15967
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

928 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:

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

Polycyclic presentation:

2^9|a^2=eh,b^2=ef,c^2=ef,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=de,e^c=ei,f^c=fi,g^d=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, 357620);