#6350994

order	512
nilpotency class	3
isoclinism	isoclinic class of #6350909
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	#537	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2545 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6258190);