#6367111

order	512
nilpotency class	3
isoclinism	isoclinic class of #6366870
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	#3482	C₂²
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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=h,e^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ehi,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

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6274307);