#397179

order	512
nilpotency class	3
isoclinism	isoclinic class of #396612
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2850
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14747
prime radical	#60469	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1412 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)(3,15,20,33)(4,21,26,40)(5,27)(6,12)(7,13)(8,14)(9,18,34,19)(10,24,37,25)(11,28)(16,31)(17,46,49,32)(22,35)(23,36,54,60)(29,39)(30,38)(41,44)(42,47,58,48)(43,61)(45,50)(51,55,56,53)(52,57)(59,63)(62,64);(1,3,7,18,8,20,6,19)(2,9,13,15,14,34,12,33)(4,22,24,45,26,44,25,16)(5,17,29,47,30,49,28,48)(10,35,21,50,37,41,40,31)(11,32,27,42,39,46,38,58)(23,52,53,63,54,62,55,43)(36,59,51,64,60,61,56,57);(1,4,6,24,8,26,7,25)(2,10,12,21,14,37,13,40)(3,16,18,44,20,45,19,22)(5,23,28,53,30,54,29,55)(9,31,15,41,34,50,33,35)(11,36,38,51,39,60,27,56)(17,43,47,62,49,63,48,52)(32,57,42,61,46,64,58,59);(1,5)(2,11)(3,17)(4,23)(6,28)(7,29)(8,30)(9,32)(10,36)(12,38)(13,27)(14,39)(15,42)(16,43)(18,47)(19,48)(20,49)(21,51)(22,52)(24,53)(25,55)(26,54)(31,57)(33,58)(34,46)(35,59)(37,60)(40,56)(41,61)(44,62)(45,63)(50,64)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 304375);