#263482

order	512
nilpotency class	3
isoclinism	isoclinic class of #263459
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂³	#535
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

729 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ehi,c^2=i,d^2=eh,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gh,d^c=di,f^c=fi,g^c=gh

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 170678);