#10529820

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	#50026	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3922 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)(3,8)(4,9)(5,10)(6,7)(11,13)(12,14)(15,16)(17,18,24,22)(19,21,25,20)(23,30,31,26)(27,28,32,29)(33,34,38,41)(35,37,40,43)(36,39,46,48)(42,45,44,47);(2,7)(3,11)(9,12)(10,16)(17,19)(18,20)(21,22)(23,27)(24,25)(26,28)(29,30)(31,32)(33,35)(34,37)(36,44)(38,40)(39,47)(41,43)(42,46)(45,48);(17,20,24,21)(18,25,22,19)(23,28,31,29)(26,32,30,27)(33,36)(34,39)(35,42)(37,45)(38,46)(40,44)(41,48)(43,47);(1,3,6,11)(2,8,7,13)(4,10,14,16)(5,12,15,9)(17,21,24,20)(18,25,22,19)(23,29,31,28)(26,32,30,27)(33,34)(35,43)(36,39)(37,40)(38,41)(42,47)(44,45)(46,48);(1,4)(2,9)(3,10)(5,8)(6,14)(7,12)(11,16)(13,15)(17,22,24,18)(19,21,25,20)(23,26,31,30)(27,28,32,29)(33,36)(34,39)(35,44)(37,47)(38,46)(40,42)(41,48)(43,45);(1,5)(2,10)(3,12)(4,13)(6,15)(7,16)(8,14)(9,11)(17,23,24,31)(18,26,22,30)(19,27,25,32)(20,28,21,29)(33,37,38,43)(34,40,41,35)(36,45,46,47)(39,44,48,42)

Polycyclic presentation:

2^9|a^2=hi,c^2=h,d^2=g,e^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,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, 10437016);