#113618

order	512
nilpotency class	3
isoclinism	isoclinic class of #113613
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	#60948	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

761 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)(3,12,16,8)(4,17,21,28)(5,11)(6,22)(7,10)(9,19,26,20)(13,30)(14,27,15,24)(18,29)(23,31)(25,32)(33,34,38,42,39,43,37,52)(35,44,46,40,47,56,53,55)(36,41,50,58,51,60,49,59)(45,57,62,64,63,61,48,54);(1,3)(2,8)(4,18)(5,15)(6,14)(7,16)(9,25)(10,12)(11,27)(13,20)(17,23)(19,32)(21,31)(22,24)(26,30)(28,29)(33,35)(34,40)(36,48)(37,53)(38,46)(39,47)(41,57)(42,56)(43,55)(44,52)(45,50)(49,63)(51,62)(54,59)(58,64)(60,61);(1,4)(2,9)(3,13)(5,20)(6,19)(7,21)(8,23)(10,26)(11,28)(12,29)(14,18)(15,31)(16,32)(17,22)(24,30)(25,27)(33,36)(34,41)(35,45)(37,49)(38,50)(39,51)(40,54)(42,58)(43,60)(44,61)(46,62)(47,63)(48,53)(52,59)(55,64)(56,57)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 20814);