#378119

order	512
nilpotency class	3
isoclinism	isoclinic class of #378118
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14207
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#3449	C₂²
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1353 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 285315);