#6434269

order	512
nilpotency class	3
isoclinism	isoclinic class of #29736
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂³	C₂³×D₄
prime radical	#23421	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1727 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2)(3,7)(4,6)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,29)(20,34)(21,26)(22,27)(23,28)(24,32)(25,36)(30,38)(31,40)(33,39)(35,42)(37,41)(43,46)(44,48)(45,47);(1,3,5,4)(2,6,8,7)(9,11)(10,13)(12,15)(14,16)(17,19,21,32,23,31,22,33)(18,24,27,29,28,39,26,40)(20,30,35,45,37,44,36,46)(25,38,41,43,42,48,34,47);(1,3,5,4)(2,7,8,6)(11,15)(13,16)(17,20,21,35,23,37,22,36)(18,25,27,41,28,42,26,34)(19,30,32,45,31,44,33,46)(24,38,29,43,39,48,40,47);(1,4,5,3)(2,6,8,7)(10,14)(13,16)(19,31)(24,39)(29,40)(30,44)(32,33)(38,48)(43,47)(45,46);(17,21,23,22)(18,26,28,27)(19,32,31,33)(20,35,37,36)(24,40,39,29)(25,34,42,41)(30,45,44,46)(38,47,48,43)

Polycyclic presentation:

2^9|b^2=f,c^2=f,d^2=fg,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,g^a=gi,c^b=ch,d^b=di

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6341465);