#6350194

order	512
nilpotency class	3
isoclinism	isoclinic class of #6350061
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₄²	C₂⁵
prime radical	#63024	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2793 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)(3,10)(4,16)(5,11)(6,12)(7,13)(8,15)(9,14)(17,18,23,27)(19,28,31,37)(20,25,22,35)(21,26,34,42)(24,36,39,29)(30,43,46,47)(32,40,33,41)(38,45,48,44);(1,3,7,4)(2,8,13,9)(5,14,6,15)(10,12,16,11)(17,19,23,31)(18,24,27,39)(20,29,22,36)(21,30,34,46)(25,37,35,28)(26,38,42,48)(32,44,33,45)(40,47,41,43);(1,3,7,4)(2,9,13,8)(5,14,6,15)(10,11,16,12)(17,20,23,22)(18,25,27,35)(19,29,31,36)(21,32,34,33)(24,37,39,28)(26,40,42,41)(30,44,46,45)(38,47,48,43);(1,4,7,3)(2,10,13,16)(5,14,6,15)(8,12,9,11)(17,21)(18,26)(19,30)(20,32)(22,33)(23,34)(24,38)(25,40)(27,42)(28,43)(29,44)(31,46)(35,41)(36,45)(37,47)(39,48);(1,5,7,6)(2,11,13,12)(3,14,4,15)(8,10,9,16)(17,20,23,22)(18,25,27,35)(19,29,31,36)(21,33,34,32)(24,37,39,28)(26,41,42,40)(30,45,46,44)(38,43,48,47)

Polycyclic presentation:

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

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, 6257390);