#10171857

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	#41116	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3907 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=h,d^2=hi,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, 10079053);