#10247496

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	#44260	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4105 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)(3,5,10,8)(4,9,14,16)(11,15,12,13)(17,18)(19,25)(20,24)(21,26)(22,23)(27,32)(28,29)(30,31)(33,34)(35,43)(36,39)(37,41)(38,46)(40,44)(42,47)(45,48);(2,7)(3,10)(9,16)(11,12)(17,19)(18,21)(20,28)(22,26)(23,25)(24,32)(27,30)(29,31)(33,35,39,44)(34,40,36,43)(37,47,48,38)(41,46,45,42);(18,23)(19,26)(24,31)(27,28)(34,36)(40,43)(41,45)(42,46);(1,3,6,10)(2,8,7,5)(4,12,14,11)(9,15,16,13)(17,19)(18,21)(20,27)(22,26)(23,25)(24,29)(28,30)(31,32)(33,36,39,34)(35,40,44,43)(37,41,48,45)(38,46,47,42);(1,4,6,14)(2,9,7,16)(3,11,10,12)(5,15,8,13)(17,20,22,30)(18,24,23,31)(19,27,26,28)(21,29,25,32)(33,37,39,48)(34,41,36,45)(35,38,44,47)(40,42,43,46);(1,5,6,8)(2,10,7,3)(4,13,14,15)(9,11,16,12)(17,21)(18,19)(20,29)(22,25)(23,26)(24,27)(28,31)(30,32)(33,38)(34,42)(35,37)(36,46)(39,47)(40,41)(43,45)(44,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,d^2=gi,e^2=g,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, 10154692);