#232846

order	512
nilpotency class	3
isoclinism	isoclinic class of #232834
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1105 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)(3,17)(4,20)(5,12)(6,34)(7,40)(8,18)(9,15)(10,16)(11,25)(13,38)(14,37)(19,26)(21,60)(22,61)(23,53)(24,36)(27,35)(28,59)(29,56)(30,49)(31,54)(32,51)(33,52)(39,64)(41,43)(42,58)(44,50)(45,55)(46,62)(47,48)(57,63);(1,3,9,23)(2,4,15,27)(5,29,32,19)(6,35,38,20)(7,41,33,21)(8,11,44,47)(10,24,30,57)(12,48,51,25)(13,53,34,17)(14,43,55,60)(16,28,49,62)(18,56,50,26)(22,37,58,45)(31,59,39,46)(36,54,63,64)(40,61,52,42);(1,4,9,27)(2,11,15,47)(3,18,23,50)(5,25,32,48)(6,36,38,63)(7,42,33,61)(8,43,44,60)(10,28,30,62)(12,46,51,59)(13,21,34,41)(14,24,55,57)(16,22,49,58)(17,31,53,39)(19,54,29,64)(20,52,35,40)(26,37,56,45);(1,5,30,7)(2,12,49,40)(3,19,57,21)(4,25,62,42)(6,8,31,37)(9,32,10,33)(11,46,58,20)(13,50,64,55)(14,34,18,54)(15,51,16,52)(17,26,63,60)(22,35,47,59)(23,29,24,41)(27,48,28,61)(36,43,53,56)(38,44,39,45)

Polycyclic presentation:

2^9|b^2=h,c^2=h,d^2=hi,e^2=h,f^2=hi,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fhi,g^a=ghi,c^b=cg,d^b=dh,e^b=eh,f^b=fh,g^b=gi,e^c=ei,f^c=fhi,g^c=gi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 140042);