#6365570

order	512
nilpotency class	3
isoclinism	isoclinic class of #6365522
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₄²	C₂⁵
prime radical	#3506	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1466 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 64 points:

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

Polycyclic presentation:

2^9|b^2=h,c^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fh,g^a=gi,c^b=ch,d^b=df,e^b=ei,f^b=fh,g^c=gi,f^d=fh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306799, #72306801, #72306829
maximal normal subgroup of	#72306799, #72306801, #72306829

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6272766);