#237278

order	512
nilpotency class	3
isoclinism	isoclinic class of #237257
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1108 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^2=i,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^a=gi,c^b=cg,e^b=eh,g^b=gi,d^c=dhi,e^c=ei,f^c=fi,g^c=gi

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, 144474);