#244542

order	512
nilpotency class	3
isoclinism	isoclinic class of #244523
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	C₂³×C₄	C₂⁴
prime radical	#61776	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

956 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)(3,7)(4,8)(5,9)(6,10)(11,17)(12,18)(13,19)(14,27)(15,22)(16,23)(20,25)(21,28)(24,31)(26,30)(29,32)(34,40)(35,45)(36,50)(37,38)(43,44)(46,55)(47,61)(48,49)(51,52)(53,57)(54,60)(63,64);(1,3,6,13)(2,7,10,19)(4,11,15,25)(5,12,16,26)(8,20,22,17)(9,18,23,30)(14,24,28,32)(21,31,27,29)(33,34)(35,46)(36,42)(37,43)(38,40)(39,44)(41,45)(47,60)(48,56)(49,55)(50,53)(51,57)(52,58)(54,59)(61,64)(62,63);(1,4)(2,8)(3,11)(5,14)(6,15)(7,17)(9,21)(10,22)(12,24)(13,25)(16,28)(18,29)(19,20)(23,27)(26,32)(30,31)(33,35,39,49)(34,41,44,56)(36,47,52,61)(37,45,38,48)(40,46,43,55)(42,54,58,63)(50,62,51,59)(53,60,57,64);(1,5)(2,9)(3,12)(4,14)(6,16)(7,18)(8,21)(10,23)(11,24)(13,26)(15,28)(17,29)(19,30)(20,31)(22,27)(25,32)(33,36)(34,42)(35,47)(37,51)(38,50)(39,52)(40,53)(41,54)(43,57)(44,58)(45,59)(46,60)(48,62)(49,61)(55,64)(56,63)

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 151738);