#6349655

order	512
nilpotency class	3
isoclinism	isoclinic class of #6349612
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	#62835	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2770 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)(4,9)(5,10)(6,11)(7,12)(8,14)(15,16)(17,18,23,26)(19,29,33,40)(20,34,35,41)(21,28,22,27)(24,32,39,31)(25,37,42,36)(30,44,45,38)(43,47,48,46);(1,3,7,4)(2,8,12,16)(5,14,6,15)(9,11,13,10)(17,19,23,33)(18,24,26,39)(20,30,35,45)(21,32,22,31)(25,38,42,44)(27,40,28,29)(34,48,41,43)(36,47,37,46);(1,4,7,3)(2,9,12,13)(5,14,6,15)(8,11,16,10)(17,20,23,35)(18,25,26,42)(19,30,33,45)(21,36,22,37)(24,38,39,44)(27,34,28,41)(29,43,40,48)(31,46,32,47);(1,5,7,6)(2,10,12,11)(3,14,4,15)(8,9,16,13)(18,26)(20,35)(24,39)(27,28)(29,40)(30,45)(36,37)(46,47);(1,5,7,6)(2,11,12,10)(3,15,4,14)(8,9,16,13)(17,21,23,22)(18,27,26,28)(19,31,33,32)(20,36,35,37)(24,29,39,40)(25,34,42,41)(30,46,45,47)(38,43,44,48)

Polycyclic presentation:

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

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, 6256851);