#244569

order	512
nilpotency class	3
isoclinism	isoclinic class of #244547
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#62514	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

925 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 151765);