#8550920

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3751 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=g,d^2=gh,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8458116);