#6368608

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
prime radical	#19227	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1453 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)(3,11)(4,8)(5,18)(6,9)(7,13)(10,21)(12,29)(14,27)(15,32)(16,28)(17,25)(19,23)(20,26)(22,31)(24,30)(33,34,40,44)(35,47,50,61)(36,52,37,43)(38,46,39,45)(41,56,58,51)(42,55,59,53)(48,57,54,64)(49,62,63,60);(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)(34,41,44,58)(36,48,37,54)(38,56,39,51)(42,57,59,64)(43,60,52,62)(45,61,46,47)(49,53,63,55);(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)(34,42,44,59)(35,48,50,54)(38,53,39,55)(41,57,58,64)(43,45,52,46)(47,60,61,62)(49,56,63,51);(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)(34,43,44,52)(35,49,50,63)(38,53,39,55)(41,57,58,64)(42,45,59,46)(47,62,61,60)(48,56,54,51);(2,9)(7,23)(8,25)(10,18)(11,27)(22,32)(24,29)(26,28)(34,44)(35,50)(36,37)(45,46)(51,56)(53,55)(57,64)(60,62)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=i,d^2=h,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, #72306657, #72306675
maximal normal subgroup of	#72306633, #72306657, #72306675

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6275804);