#8459059

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	#55356	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4087 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)(3,10)(4,7)(5,12)(6,8)(9,14)(11,15)(13,16)(20,28)(24,31)(27,32)(29,30)(33,34)(35,43)(36,40)(37,45)(38,41)(39,42)(44,47)(46,48);(1,3,7,6)(2,8,4,10)(5,11,14,13)(9,15,12,16)(17,18)(19,23)(20,24)(21,25)(22,26)(27,30)(28,31)(29,32)(33,35)(34,37)(36,44)(38,40)(39,45)(41,46)(42,43)(47,48);(1,2,7,4)(3,10,6,8)(5,9,14,12)(11,16,13,15)(17,19)(18,21)(20,27)(22,25)(23,26)(24,29)(28,32)(30,31)(33,36,39,46)(34,40,42,48)(35,44,45,41)(37,38,43,47);(1,4,7,2)(3,8,6,10)(5,12,14,9)(11,15,13,16)(17,19)(18,23)(20,27)(21,26)(22,25)(24,30)(28,32)(29,31)(33,37)(34,35)(36,38)(39,43)(40,44)(41,48)(42,45)(46,47);(1,5)(2,9)(3,11)(4,12)(6,13)(7,14)(8,15)(10,16)(17,20,22,28)(18,24,26,31)(19,27,25,32)(21,29,23,30)(33,35)(34,37)(36,44)(38,40)(39,45)(41,46)(42,43)(47,48);(1,6,7,3)(2,8,4,10)(5,13,14,11)(9,15,12,16)(17,21,22,23)(18,25,26,19)(20,29,28,30)(24,32,31,27)(33,38)(34,41)(35,40)(36,43)(37,46)(39,47)(42,44)(45,48)

Polycyclic presentation:

2^9|b^2=h,c^2=g,d^2=h,e^2=hi,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8366255);