#6368692

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	#62486	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1445 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=i,e^2=hi,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	#72306683, #72306699, #72306708
maximal normal subgroup of	#72306683, #72306699, #72306708

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6275888);