#10086328

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	#2388	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3896 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)(3,8,10,5)(4,9,12,15)(11,13,16,14)(17,18)(19,22)(20,27)(21,23)(24,28)(25,31)(26,29)(30,32)(34,39)(35,41)(37,46)(43,47);(2,7)(3,10)(9,15)(11,16)(17,19,21,26)(18,22,23,29)(20,25,28,32)(24,30,27,31)(34,39)(35,41)(36,45)(44,48);(4,12)(9,15)(11,16)(13,14)(18,23)(22,29)(24,27)(30,31)(33,34,38,39)(35,42,41,40)(36,46,45,37)(43,44,47,48);(1,3)(2,8)(4,11)(5,7)(6,10)(9,13)(12,16)(14,15)(17,18)(19,22)(20,24)(21,23)(25,30)(26,29)(27,28)(31,32)(33,35,38,41)(34,40,39,42)(36,43,45,47)(37,44,46,48);(1,4,6,12)(2,9,7,15)(3,11,10,16)(5,14,8,13)(17,19)(18,22)(20,25)(21,26)(23,29)(24,30)(27,31)(28,32)(33,36,38,45)(34,37,39,46)(35,43,41,47)(40,44,42,48);(1,5,6,8)(2,3,7,10)(4,13,12,14)(9,16,15,11)(17,20,21,28)(18,24,23,27)(19,25,26,32)(22,30,29,31)(33,37)(34,36)(35,44)(38,46)(39,45)(40,43)(41,48)(42,47)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9993524);