#440845

order	512
nilpotency class	3
isoclinism	isoclinic class of #427311
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2471
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15669
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1029 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)(3,14,18,9)(4,19,24,38)(5,25,27,41)(6,13)(7,30)(8,12)(10,22,37,23)(11,26,40,29)(15,43)(16,47)(17,34,28,33)(20,42)(21,53)(31,45)(32,48)(35,46)(36,56)(39,49)(44,59,64,62)(50,57)(51,54)(52,61,63,60)(55,58);(1,3,7,17,8,18,6,28)(2,9,30,33,12,14,13,34)(4,20,22,46,24,45,23,15)(5,16,26,49,27,50,29,48)(10,35,19,31,37,43,38,42)(11,39,25,57,40,32,41,47)(21,52,55,64,56,63,54,44)(36,61,51,62,53,60,58,59);(1,4)(2,10,12,37)(3,15)(5,21)(6,22)(7,23)(8,24)(9,31,14,42)(11,36,40,53)(13,19,30,38)(16,44)(17,45)(18,46)(20,28)(25,58,41,51)(26,54)(27,56)(29,55)(32,59,39,62)(33,35,34,43)(47,60,57,61)(48,52)(49,63)(50,64);(1,5)(2,11)(3,16)(4,21)(6,29)(7,26)(8,27)(9,32)(10,36)(12,40)(13,41)(14,39)(15,44)(17,49)(18,50)(19,51)(20,52)(22,55)(23,54)(24,56)(25,30)(28,48)(31,59)(33,47)(34,57)(35,60)(37,53)(38,58)(42,62)(43,61)(45,63)(46,64)

Polycyclic presentation:

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

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, 348041);