#6368675

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

1462 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:

(1,2,8,13)(3,14,18,29)(4,5,10,22)(6,12,7,11)(9,24,27,17)(15,30,28,16)(19,25,23,21)(20,31,32,26)(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)(2,9)(4,15)(5,20)(6,17)(7,24)(8,18)(10,28)(11,14)(12,29)(13,27)(16,21)(19,31)(22,32)(23,26)(25,30)(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);(1,4,8,10)(2,5,13,22)(3,15,18,28)(6,23,7,19)(9,20,27,32)(11,25,12,21)(14,30,29,16)(17,26,24,31)(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);(1,5)(2,10)(3,16)(4,13)(6,21)(7,25)(8,22)(9,26)(11,23)(12,19)(14,15)(17,32)(18,30)(20,24)(27,31)(28,29)(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);(1,6,8,7)(2,11,13,12)(3,17,18,24)(4,19,10,23)(5,21,22,25)(9,14,27,29)(15,31,28,26)(16,32,30,20)(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=i,c^2=h,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	#72306688, #72306696, #72306707
maximal normal subgroup of	#72306688, #72306696, #72306707

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6275871);