#10171834

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	#53552	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3899 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=gh,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, 10079030);