#146776

order	512
nilpotency class	4
isoclinism	isoclinic class of #13033
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1879
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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=hi,e^2=hi,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=dhi,e^b=eh,h^b=hi,d^c=dh,e^c=eh,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72314212
maximal normal subgroup of	#72314212

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53972);