#8583464

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	#2410	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4176 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=gh,d^2=g,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, 8490660);