#9230882

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	#2669	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4016 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=g,d^2=gh,e^2=hi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^d=egi,f^d=fgi,f^e=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, 9138078);