#10482491

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	#56349	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3942 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,d^2=i,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10389687);