#232833

order	512
nilpotency class	3
isoclinism	isoclinic class of #232820
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	#60961	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1116 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)(3,18)(4,21)(5,14)(6,35)(7,40)(8,43)(9,16)(10,17)(11,26)(12,33)(13,39)(15,54)(19,31)(20,27)(22,45)(23,42)(24,49)(25,37)(28,36)(29,46)(30,48)(32,59)(34,50)(38,56)(41,63)(44,52)(47,62)(51,57)(53,61)(55,64)(58,60);(1,3)(2,4)(5,20)(6,36)(7,41)(8,44)(9,24)(10,25)(11,31)(12,26)(13,49)(14,52)(15,45)(16,28)(17,29)(18,35)(19,27)(21,39)(22,34)(23,38)(30,55)(32,53)(33,60)(37,59)(40,62)(42,50)(43,58)(46,57)(47,54)(48,61)(51,64)(56,63);(1,4,9,28)(2,11,16,44)(3,19,24,43)(5,26,33,52)(6,37,39,64)(7,42,34,62)(8,45,31,63)(10,29,30,61)(12,46,14,53)(13,22,35,41)(15,55,56,25)(17,47,48,23)(18,57,49,32)(20,59,60,51)(21,50,36,40)(27,54,58,38);(1,5,30,7)(2,12,48,50)(3,20,55,41)(4,26,61,42)(6,31,57,38)(8,32,54,39)(9,33,10,34)(11,46,23,36)(13,19,59,56)(14,17,40,16)(15,35,43,51)(18,58,64,45)(21,44,53,47)(22,24,60,25)(27,37,63,49)(28,52,29,62)

Polycyclic presentation:

2^9|c^2=h,d^2=hi,e^2=h,f^2=hi,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fhi,g^a=ghi,c^b=cg,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, 140029);