#6369190

order	512
nilpotency class	3
isoclinism	isoclinic class of #6369150
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	#3169	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1451 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306663, #72306683, #72306720
maximal normal subgroup of	#72306663, #72306683, #72306720

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6276386);