#6526815

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500474
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#60559	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1661 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6434011);