#253404

order	512
nilpotency class	3
isoclinism	isoclinic class of #253387
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

688 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=i,d^2=hi,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=ghi,c^b=cg,d^b=dh,f^b=fi,d^c=di,e^c=eh,f^c=fi,g^c=gh,f^d=fh

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, 160600);