#6351969

order	512
nilpotency class	3
isoclinism	isoclinic class of #6351885
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	#63051	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2671 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)(3,13)(4,14)(5,10)(6,11)(7,12)(8,15)(9,16)(17,18,24,28)(19,29,33,40)(20,26,21,35)(22,27,23,37)(25,38,41,32)(30,45,31,44)(34,42,36,43)(39,46,48,47);(1,3,7,9)(2,8,12,4)(5,14,6,15)(10,13,11,16)(17,19,24,33)(18,25,28,41)(20,30,21,31)(22,32,23,38)(26,39,35,48)(27,40,37,29)(34,46,36,47)(42,45,43,44);(1,4,7,8)(2,9,12,3)(5,16,6,13)(10,15,11,14)(17,20,24,21)(18,26,28,35)(19,30,33,31)(22,34,23,36)(25,39,41,48)(27,42,37,43)(29,44,40,45)(32,46,38,47);(1,2)(3,8)(4,9)(5,11)(6,10)(7,12)(13,15)(14,16)(17,21,24,20)(18,26,28,35)(19,31,33,30)(22,34,23,36)(25,39,41,48)(27,43,37,42)(29,45,40,44)(32,46,38,47);(1,5,7,6)(2,10,12,11)(3,14,9,15)(4,16,8,13)(17,22,24,23)(18,27,28,37)(19,32,33,38)(20,34,21,36)(25,40,41,29)(26,42,35,43)(30,46,31,47)(39,45,48,44)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6259165);