#392946

order	512
nilpotency class	3
isoclinism	isoclinic class of #32950
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2847
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14697
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1347 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=f,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,f^a=fi,g^a=gi,e^b=ei,g^d=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72281077, #72281094, #72281111
maximal normal subgroup of	#72281077, #72281094, #72281111

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 300142);