#113818

order	512
nilpotency class	3
isoclinism	isoclinic class of #113815
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	C₂⁴	#127
LCS	G ▷ C₂×C₄² ▷ 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)(3,12,15,8)(4,16)(5,21)(6,11)(7,10)(9,18)(13,26,30,32)(14,24,22,25)(17,29,31,23)(19,27)(20,28)(33,34)(35,45,48,59)(36,49,52,57)(37,42)(38,43)(39,44)(40,47,55,53)(41,51,58,50)(46,54)(56,62)(60,63)(61,64);(1,3,7,15)(2,8,10,12)(4,17,20,31)(5,14,6,22)(9,26,27,32)(11,24,21,25)(13,18,30,19)(16,23,28,29)(33,35)(34,40)(36,46)(37,53)(38,47)(39,48)(41,56)(42,59)(43,45)(44,55)(49,60)(50,63)(51,61)(52,62)(54,58)(57,64);(1,4,7,20)(2,9,10,27)(3,13,15,30)(5,19,6,18)(8,23,12,29)(11,28,21,16)(14,31,22,17)(24,26,25,32)(33,36)(34,41)(35,46)(37,51)(38,50)(39,52)(40,54)(42,49)(43,57)(44,58)(45,60)(47,63)(48,62)(53,61)(55,56)(59,64)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 21014);