#430347

order	512
nilpotency class	3
isoclinism	isoclinic class of #418498
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2673
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15418
prime radical	#23414	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1152 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,d^2=ef,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,e^c=ei,f^c=fi,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, 337543);