#9230892

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	#2682	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3951 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:

(1,2,8,6)(3,12,15,9)(4,10,5,14)(7,11,16,13)(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)(2,9)(4,13)(5,11)(6,12)(7,14)(8,15)(10,16)(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)(2,10,6,14)(3,13,15,11)(7,9,16,12)(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)(2,10,6,14)(3,11,15,13)(7,9,16,12)(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);(1,6,8,2)(3,12,15,9)(4,14,5,10)(7,11,16,13)(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);(1,7,8,16)(2,11,6,13)(3,14,15,10)(4,9,5,12)(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=h,c^2=g,d^2=gh,e^2=ghi,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, 9138088);