#113733

order	512
nilpotency class	3
isoclinism	isoclinic class of #113733
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	#60961	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

766 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:

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

Polycyclic presentation:

2^9|a^2=g,d^2=i,e^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fhi,c^b=cf,d^b=di,f^b=fh,g^b=ghi,d^c=dhi,e^c=ei,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, 20929);