#113839

order	512
nilpotency class	3
isoclinism	isoclinic class of #113838
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

767 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=i,e^2=hi,f^2=h,b^a=bd,c^a=ce,d^a=dh,f^a=fhi,c^b=cf,d^b=di,e^b=eh,f^b=fh,g^b=ghi,d^c=di,e^c=ehi,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, 21035);