#7173510

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996405
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#61200	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1678 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)(3,6,9,13)(4,11,12,7)(10,16,15,14)(17,18,24,30)(19,31,35,45)(20,36,39,42)(21,27,32,47)(22,29,23,28)(25,37,43,34)(26,38,44,33)(40,48,41,46);(3,9)(4,12)(6,13)(7,11)(17,19,24,35)(18,25,30,43)(20,32,39,21)(22,37,23,34)(26,27,44,47)(28,45,29,31)(33,41,38,40)(36,46,42,48);(1,3)(2,6)(4,10)(5,9)(7,14)(8,13)(11,16)(12,15)(17,20,22,38,24,39,23,33)(18,26,28,42,30,44,29,36)(19,32,37,40,35,21,34,41)(25,27,45,48,43,47,31,46);(3,9)(6,13)(10,15)(14,16)(17,21)(18,27)(19,33)(20,37)(22,41)(23,40)(24,32)(25,42)(26,31)(28,48)(29,46)(30,47)(34,39)(35,38)(36,43)(44,45);(1,4,5,12)(2,7,8,11)(3,10,9,15)(6,14,13,16)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7080706);