#122105

order	512
nilpotency class	3
isoclinism	isoclinic class of #122103
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1446
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

554 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=f,d^2=h,e^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fhi,c^b=cf,d^b=di,e^b=eh,d^c=di,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, 29301);