#6350588

order	512
nilpotency class	3
isoclinism	isoclinic class of #6350525
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	#63009	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2739 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,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,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, 6257784);