#237232

order	512
nilpotency class	3
isoclinism	isoclinic class of #237213
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	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1093 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)(41,52)(45,58)(46,55)(47,57)(49,54)(50,59)(51,60)(53,56)(61,63)(62,64);(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)(45,50)(48,63)(51,61)(52,62)(54,60)(59,64);(1,4,6,15)(2,8,10,22)(3,12,14,26)(5,16,19,28)(7,20,21,31)(9,23,17,29)(11,25,24,32)(13,27,18,30)(33,36)(34,41)(35,45)(37,50)(38,52)(39,51)(40,54)(42,48)(43,60)(44,59)(46,61)(47,62)(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)(41,48,59,60)(45,61,62,49)(54,58,63,64)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^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=dh,e^c=ei,f^c=fi,g^c=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 144428);