#6350434

order	512
nilpotency class	3
isoclinism	isoclinic class of #6350397
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	#63001	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2727 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6257630);