#245312

order	512
nilpotency class	3
isoclinism	isoclinic class of #245303
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	#18946	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

938 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|b^2=h,d^2=i,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=gi,c^b=cg,e^b=eh,g^b=gh,e^c=ei,f^c=fi,g^c=gh,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, 152508);