#244572

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

950 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^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, 151768);