#6368669

order	512
nilpotency class	3
isoclinism	isoclinic class of #6368558
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
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1470 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)(33,34,39,44)(35,40,47,57)(36,42,49,58)(37,43,38,52)(41,51,59,48)(45,54,62,64)(46,55,50,56)(53,60,63,61);(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)(33,35,39,47)(34,40,44,57)(36,45,49,62)(37,50,38,46)(41,53,59,63)(42,54,58,64)(43,56,52,55)(48,61,51,60);(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)(34,41)(35,45)(37,48)(38,51)(39,49)(40,53)(42,52)(43,58)(44,59)(46,60)(47,62)(50,61)(54,55)(56,64)(57,63);(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)(33,36)(34,42)(35,45)(37,51)(38,48)(39,49)(40,54)(41,52)(43,59)(44,58)(46,61)(47,62)(50,60)(53,55)(56,63)(57,64);(2,9)(7,23)(8,25)(10,18)(11,27)(22,32)(24,29)(26,28)(33,37,39,38)(34,43,44,52)(35,46,47,50)(36,48,49,51)(40,55,57,56)(41,58,59,42)(45,60,62,61)(53,54,63,64)

Polycyclic presentation:

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

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306633, #72306654, #72306678
maximal normal subgroup of	#72306633, #72306654, #72306678

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6275865);