#8445958

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	#40361	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4024 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)(3,9)(8,13)(10,15)(17,18,24,27)(19,25,22,23)(20,30,21,26)(28,31,29,32)(33,34,38,42)(35,43,44,39)(36,40,45,48)(37,41,47,46);(1,2,5,6)(3,7,9,12)(4,8,11,13)(10,14,15,16)(17,19,24,22)(18,25,27,23)(20,29,21,28)(26,31,30,32)(33,35)(34,39)(36,41)(37,40)(38,44)(42,43)(45,46)(47,48);(1,2)(3,7)(4,8)(5,6)(9,12)(10,14)(11,13)(15,16)(17,20,24,21)(18,26,27,30)(19,28,22,29)(23,31,25,32)(33,34)(35,39)(36,40)(37,41)(38,42)(43,44)(45,48)(46,47);(1,2)(3,7)(4,8)(5,6)(9,12)(10,14)(11,13)(15,16)(17,21,24,20)(18,26,27,30)(19,29,22,28)(23,31,25,32)(33,34)(35,43)(36,40)(37,46)(38,42)(39,44)(41,47)(45,48);(1,3)(2,7)(4,10)(5,9)(6,12)(8,14)(11,15)(13,16)(17,22,24,19)(18,23,27,25)(20,29,21,28)(26,31,30,32)(33,36,38,45)(34,40,42,48)(35,41,44,46)(37,43,47,39);(1,4,5,11)(2,8,6,13)(3,10,9,15)(7,14,12,16)(17,23)(18,19)(20,31)(21,32)(22,27)(24,25)(26,28)(29,30)(33,37,38,47)(34,41,42,46)(35,40,44,48)(36,43,45,39)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=gh,d^2=gh,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8353154);