#114630

order	512
nilpotency class	3
isoclinism	isoclinic class of #114623
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	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

745 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=i,e^2=h,f^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fhi,c^b=cf,d^b=di,f^b=fi,g^b=ghi,d^c=dhi,e^c=eh,f^c=fi

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, 21826);