#6369120

order	512
nilpotency class	3
isoclinism	isoclinic class of #6369070
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

1459 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)(3,11,15,27)(4,8,18,25)(5,19,22,10)(7,20,12,14)(13,29,31,24)(16,30,32,23)(17,28,21,26)(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);(1,3,6,15)(2,7,9,12)(4,16,18,32)(5,20,22,14)(8,24,25,29)(10,27,19,11)(13,21,31,17)(23,28,30,26)(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)(3,12)(4,8)(5,10)(6,9)(7,15)(11,20)(13,30)(14,27)(16,29)(17,26)(18,25)(19,22)(21,28)(23,31)(24,32)(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);(1,4,6,18)(2,8,9,25)(3,13,15,31)(5,21,22,17)(7,23,12,30)(10,28,19,26)(11,29,27,24)(14,32,20,16)(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)(7,12)(8,25)(10,19)(11,27)(23,30)(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=ch,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

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306667, #72306687, #72306720
maximal normal subgroup of	#72306667, #72306687, #72306720

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6276316);