#128077

order	512
nilpotency class	4
isoclinism	isoclinic class of #128061
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#8101
commutator	C₂⁴	C₂×C₄²
Frattini, cosocle	C₂²×#41	C₂³
prime radical	C₂²×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂⁴ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂²×2₋¹⁺⁴ ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂³ ▷ C₂ ▷ C₁

1259 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)(3,13,16,33)(4,18,20,9)(5,22,24,10)(6,26,27,11)(7,12)(8,15)(14,37)(17,46)(19,31)(21,32)(23,34)(25,35)(28,39,47,42)(29,43,53,63)(30,44)(36,52)(38,45)(40,60,61,58)(41,54)(48,50)(49,57,51,56)(55,59)(62,64);(1,3,6,17,7,16,27,45)(2,8,11,29,12,30,26,53)(4,14,21,40,20,41,50,61)(5,15,25,43,24,44,52,63)(9,28,32,55,18,47,48,64)(10,33,35,46,22,13,36,38)(19,39,49,59,23,42,51,62)(31,37,56,58,34,54,57,60);(1,4)(2,9)(3,14)(5,23)(6,21)(7,20)(8,28)(10,34)(11,32)(12,18)(13,37)(15,42)(16,41)(17,40)(19,24)(22,31)(25,51)(26,48)(27,50)(29,55)(30,47)(33,54)(35,57)(36,56)(38,58)(39,44)(43,62)(45,61)(46,60)(49,52)(53,64)(59,63)

Polycyclic presentation:

2^9|a^2=e,b^2=f,f^2=i,b^a=bd,c^a=cg,d^a=dg,f^a=fh,g^a=gi,h^a=hi,d^b=dh,e^b=eg,g^b=gi,d^c=di,e^c=ei,e^d=ei,f^e=fi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72302902, #72302905, #72302912
maximal normal subgroup of	#72302902, #72302905, #72302912

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 35273);