#8588962

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	#2416	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4051 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=h,d^2=h,e^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,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, 8496158);