#421075

order	512
nilpotency class	3
isoclinism	isoclinic class of #421074
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×#3020	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1805 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)(3,13)(4,18)(5,10)(6,26)(7,12)(8,16)(9,21)(11,30)(14,43)(15,36)(17,33)(19,50)(20,52)(22,40)(23,32)(24,41)(25,38)(27,39)(28,35)(29,37)(31,47)(34,53)(42,45)(44,62)(46,57)(48,58)(49,54)(51,61)(55,60)(56,64)(59,63);(1,3,6,27,7,16,30,17)(2,8,11,39,12,13,26,33)(4,19,21,45,22,14,28,47)(5,23,24,46,25,15,29,48)(9,31,18,43,35,42,40,50)(10,36,37,57,38,32,41,58)(20,44,53,64,54,51,55,63)(34,59,52,61,60,56,49,62);(1,4)(2,9)(3,14)(5,20)(6,28)(7,22)(8,31)(10,34)(11,40)(12,35)(13,42)(15,44)(16,19)(17,47)(18,26)(21,30)(23,51)(24,55)(25,54)(27,45)(29,53)(32,56)(33,43)(36,59)(37,49)(38,60)(39,50)(41,52)(46,64)(48,63)(57,62)(58,61);(1,5,7,25)(2,10,12,38)(3,15,16,23)(4,20,22,54)(6,29,30,24)(8,32,13,36)(9,34,35,60)(11,41,26,37)(14,44,19,51)(17,48,27,46)(18,49,40,52)(21,55,28,53)(31,56,42,59)(33,58,39,57)(43,61,50,62)(45,64,47,63)

Polycyclic presentation:

2^9|b^2=ef,d^2=i,f^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,g^b=gi,f^c=fi,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, 328271);