#6436828

order	512
nilpotency class	3
isoclinism	isoclinic class of #6433075
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#29751	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1702 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|b^2=f,d^2=i,e^2=h,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,e^b=ei,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6344024);