#10247661

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	#44328	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) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,d^2=hi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^d=egi,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, 10154857);