#6358004

order	512
nilpotency class	3
isoclinism	isoclinic class of #6357957
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#127
commutator, Frattini, cosocle	C₄²	C₂⁵
prime radical	#61011	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1817 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=fh,c^2=gh,d^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,e^b=ei,d^c=di,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, 6265200);