#115376

order	512
nilpotency class	3
isoclinism	isoclinic class of #111483
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1438
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#121	C₂×C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

751 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=h,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,c^b=cf,d^b=dh,e^b=ei,f^b=fh,d^c=di,f^c=fh,g^c=ghi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 22572);