#146762

order	512
nilpotency class	4
isoclinism	isoclinic class of #12948
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1864
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#10788
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

939 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72314263, #72314275
maximal normal subgroup of	#72314263, #72314275

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53958);