#6364505

order	512
nilpotency class	3
isoclinism	isoclinic class of #6364250
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

1508 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)(3,13,17,22)(4,9,6,11)(5,10,20,28)(8,16,24,14)(15,29,31,26)(18,25,21,27)(19,32,30,23)(33,34,39,43)(35,40,46,55)(36,48,50,56)(37,42,38,51)(41,52,57,49)(44,58,60,63)(45,54,47,61)(53,62,64,59);(1,3,7,17)(2,8,12,24)(4,14,6,16)(5,19,20,30)(9,22,11,13)(10,26,28,29)(15,21,31,18)(23,25,32,27)(33,35)(34,40)(36,44)(37,45)(38,47)(39,46)(41,53)(42,54)(43,55)(48,58)(49,59)(50,60)(51,61)(52,62)(56,63)(57,64);(1,4,7,6)(2,9,12,11)(3,14,17,16)(5,18,20,21)(8,22,24,13)(10,25,28,27)(15,30,31,19)(23,26,32,29)(33,36,39,50)(34,41,43,57)(35,44,46,60)(37,49,38,52)(40,53,55,64)(42,56,51,48)(45,59,47,62)(54,63,61,58);(1,5,7,20)(2,10,12,28)(3,15,17,31)(4,18,6,21)(8,23,24,32)(9,25,11,27)(13,29,22,26)(14,30,16,19)(33,37,39,38)(34,42,43,51)(35,45,46,47)(36,49,50,52)(40,54,55,61)(41,56,57,48)(44,59,60,62)(53,63,64,58);(1,4,7,6)(2,11,12,9)(3,16,17,14)(5,18,20,21)(8,22,24,13)(10,27,28,25)(15,19,31,30)(23,26,32,29)(35,46)(40,55)(44,60)(45,47)(53,64)(54,61)(58,63)(59,62)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=hi,d^2=hi,e^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=ehi,f^b=fh,g^c=gi,f^d=fh

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306952, #72307001, #72307028
maximal normal subgroup of	#72306952, #72307001, #72307028

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6271701);