#394343

order	512
nilpotency class	3
isoclinism	isoclinic class of #394343
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14716
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#60387	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2514 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)(3,15)(4,21)(5,11)(6,13)(7,12)(8,14)(9,18)(10,25)(16,42)(17,41)(19,33)(20,32)(22,30)(23,51)(24,36)(26,35)(27,39)(28,38)(29,37)(31,46)(34,52)(40,45)(43,55)(44,56)(47,58)(48,57)(49,53)(50,61)(54,59)(60,62)(63,64);(1,3,6,18,8,20,7,19)(2,9,12,15,14,33,13,32)(4,22,24,45,26,16,25,44)(5,17,27,46,28,47,29,48)(10,30,35,56,36,42,21,40)(11,31,37,41,38,57,39,58)(23,50,52,63,53,43,54,62)(34,55,51,64,59,61,49,60);(1,4)(2,10)(3,16)(5,23)(6,25)(7,24)(8,26)(9,30)(11,34)(12,21)(13,35)(14,36)(15,40)(17,43)(18,45)(19,44)(20,22)(27,54)(28,53)(29,52)(31,55)(32,56)(33,42)(37,49)(38,59)(39,51)(41,60)(46,63)(47,50)(48,62)(57,61)(58,64);(1,5)(2,11)(3,17)(4,23)(6,27)(7,29)(8,28)(9,31)(10,34)(12,37)(13,39)(14,38)(15,41)(16,43)(18,46)(19,48)(20,47)(21,49)(22,50)(24,52)(25,54)(26,53)(30,55)(32,58)(33,57)(35,51)(36,59)(40,60)(42,61)(44,62)(45,63)(56,64)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72341337, #72341340, #72341344, …
maximal normal subgroup of	#72341337, #72341340, #72341344, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 301539);