#10482486

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	#58656	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3904 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)(3,4)(5,8)(6,9)(7,10)(11,12)(13,15)(14,16)(17,18,24,26)(19,27,22,23)(20,29,21,25)(28,31,30,32)(36,44)(39,45)(42,48)(46,47);(1,3,7,5)(2,8,10,4)(6,11,14,13)(9,15,16,12)(17,19,24,22)(18,23,26,27)(20,28,21,30)(25,31,29,32)(33,34)(35,40)(36,45)(37,41)(38,43)(39,44)(42,46)(47,48);(17,20,24,21)(18,25,26,29)(19,28,22,30)(23,31,27,32)(33,35)(34,38)(36,42)(37,43)(39,46)(40,41)(44,48)(45,47);(1,4,7,8)(2,3,10,5)(6,12,14,15)(9,11,16,13)(17,21,24,20)(18,25,26,29)(19,28,22,30)(23,32,27,31)(33,36,37,44)(34,39,41,45)(35,42,43,48)(38,46,40,47);(1,5,7,3)(2,8,10,4)(6,11,14,13)(9,12,16,15)(17,22,24,19)(18,23,26,27)(20,30,21,28)(25,31,29,32)(33,35)(34,40)(36,42)(37,43)(38,41)(39,47)(44,48)(45,46);(1,6,7,14)(2,9,10,16)(3,11,5,13)(4,12,8,15)(17,23)(18,22)(19,26)(20,31)(21,32)(24,27)(25,30)(28,29)(33,34)(35,38)(36,39)(37,41)(40,43)(42,46)(44,45)(47,48)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=h,d^2=gi,e^2=g,f^2=gh,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, 10389682);