#441614

order	512
nilpotency class	3
isoclinism	isoclinic class of #427241
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2471
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15687
prime radical	#3449	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:

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

Polycyclic presentation:

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

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, 348810);