#8551009

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

3818 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)(17,18,22,23)(19,26,21,24)(20,28,30,25)(27,31,29,32)(33,34,39,37)(35,40,45,46)(36,42,38,41)(43,48,44,47);(1,3,7,10)(2,8,6,5)(4,11,13,16)(9,15,14,12)(18,23)(24,26)(25,28)(31,32)(33,35)(34,40)(36,43)(37,46)(38,44)(39,45)(41,47)(42,48);(3,10)(5,8)(11,16)(12,15)(17,19,22,21)(18,24,23,26)(20,29,30,27)(25,32,28,31)(33,36,39,38)(34,41,37,42)(35,43,45,44)(40,47,46,48);(1,4)(2,9)(3,11)(5,12)(6,14)(7,13)(8,15)(10,16)(17,18)(19,24)(20,25)(21,26)(22,23)(27,31)(28,30)(29,32)(33,37,39,34)(35,40,45,46)(36,42,38,41)(43,47,44,48);(1,5)(2,3)(4,12)(6,10)(7,8)(9,11)(13,15)(14,16)(17,20,22,30)(18,25,23,28)(19,27,21,29)(24,31,26,32)(33,38,39,36)(34,42,37,41)(35,44,45,43)(40,48,46,47);(1,6,7,2)(3,5,10,8)(4,9,13,14)(11,15,16,12)(17,21,22,19)(18,24,23,26)(20,29,30,27)(25,31,28,32)

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=h,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, 8458205);