#492894

order	512
nilpotency class	4
isoclinism	isoclinic class of #33348
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂²×#99
commutator	C₂³	C₂²×C₄²
Frattini, cosocle	C₂×#41	C₂⁴
socle	C₂	C₂²×#342
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×#41 ◁ C₂×#585 ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

2241 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,f^2=i,g^2=i,b^a=be,e^a=eh,h^a=hi,f^b=fh,d^c=di,f^e=fi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72317432, #72317446, #72317456
maximal normal subgroup of	#72317432, #72317446, #72317456

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400090);