#244543

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	#61777	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:

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

Polycyclic presentation:

2^9|a^2=h,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, 151739);