#244563

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	#62602	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

952 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^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, 151759);