#10181014

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	#53693	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3895 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)(3,6)(4,9)(5,10)(7,8)(11,16)(12,14)(13,15)(17,18,22,24)(19,26,21,23)(20,25,30,29)(27,32,28,31);(1,3,7,11)(2,6,8,16)(4,12,13,5)(9,14,15,10)(17,19,22,21)(18,23,24,26)(20,28,30,27)(25,32,29,31)(33,34,39,37)(35,42,36,40)(38,41,46,45)(43,48,44,47);(2,8)(3,11)(4,13)(10,14)(18,24)(23,26)(25,29)(31,32)(33,35,39,36)(34,40,37,42)(38,43,46,44)(41,47,45,48);(1,4)(2,9)(3,12)(5,11)(6,14)(7,13)(8,15)(10,16)(17,18)(19,23)(20,29)(21,26)(22,24)(25,30)(27,32)(28,31)(33,36,39,35)(34,40,37,42)(38,44,46,43)(41,47,45,48);(1,5,7,12)(2,10,8,14)(3,13,11,4)(6,15,16,9)(17,20,22,30)(18,25,24,29)(19,27,21,28)(23,31,26,32)(33,37,39,34)(35,42,36,40)(38,41,46,45)(43,47,44,48);(1,6)(2,11)(3,8)(4,14)(5,9)(7,16)(10,13)(12,15)(17,21,22,19)(18,26,24,23)(20,27,30,28)(25,31,29,32)(33,38)(34,41)(35,43)(36,44)(37,45)(39,46)(40,47)(42,48)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=ghi,d^2=ghi,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10088210);