#10247637

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	#56034	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3841 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)(3,11,12,7)(4,9,15,14)(5,16,13,10)(17,18)(19,25)(20,22)(21,27)(23,24)(26,30)(28,29)(31,32);(1,3,6,12)(2,7,8,11)(4,13,15,5)(9,16,14,10)(18,24)(20,30)(25,27)(28,31)(33,34,38,41)(35,36,43,39)(37,40,46,48)(42,44,45,47);(2,8)(7,11)(9,14)(10,16)(17,19,23,21)(18,25,24,27)(20,28,30,31)(22,29,26,32)(33,35)(34,39)(36,41)(37,42)(38,43)(40,47)(44,48)(45,46);(1,2)(3,11)(4,14)(5,10)(6,8)(7,12)(9,15)(13,16)(17,20)(18,26)(19,28)(21,31)(22,24)(23,30)(25,32)(27,29)(33,36,38,39)(34,35,41,43)(37,44,46,47)(40,42,48,45);(1,4,6,15)(2,9,8,14)(3,5,12,13)(7,10,11,16)(17,21,23,19)(18,27,24,25)(20,28,30,31)(22,29,26,32)(33,35)(34,36)(37,45)(38,43)(39,41)(40,47)(42,46)(44,48);(1,5)(2,10)(3,4)(6,13)(7,9)(8,16)(11,14)(12,15)(17,22,23,26)(18,20,24,30)(19,29,21,32)(25,28,27,31)(33,37)(34,40)(35,42)(36,44)(38,46)(39,47)(41,48)(43,45)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=gh,d^2=i,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, 10154833);