#421073

order	512
nilpotency class	3
isoclinism	isoclinic class of #421072
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15283
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1812 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=ef,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,f^c=fi,g^c=gi,h^c=hi,f^d=fi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 328269);