#395561

order	512
nilpotency class	3
isoclinism	isoclinic class of #395560
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14725
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

2089 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72311147, #72311288, #72341342, …
maximal normal subgroup of	#72311147, #72311288, #72341342, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 302757);