#9206634

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	#2613	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3983 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)(3,10)(4,7)(5,8)(6,9)(11,16)(12,13)(14,15)(17,18)(19,25)(20,26)(21,30)(22,23)(24,29)(27,28)(31,32)(33,34,39,43)(35,37,46,40)(36,47,45,41)(38,48,44,42);(1,3,6,8)(2,5,9,10)(4,11,12,15)(7,14,13,16)(18,23)(20,30)(24,25)(27,31)(33,35,39,46)(34,40,43,37)(36,38,45,44)(41,48,47,42);(4,12)(7,13)(11,15)(14,16)(17,19,22,29)(18,24,23,25)(20,27,30,31)(21,28,26,32)(33,36,39,45)(34,41,43,47)(35,44,46,38)(37,48,40,42);(1,4,6,12)(2,7,9,13)(3,11,8,15)(5,14,10,16)(17,19)(18,25)(20,31)(21,28)(22,29)(23,24)(26,32)(27,30)(33,37)(34,35)(36,48)(38,47)(39,40)(41,44)(42,45)(43,46);(1,2)(3,5)(4,13)(6,9)(7,12)(8,10)(11,16)(14,15)(17,20,22,30)(18,26,23,21)(19,27,29,31)(24,32,25,28)(33,35)(34,37)(36,44)(38,45)(39,46)(40,43)(41,48)(42,47);(1,5)(2,8)(3,9)(4,14)(6,10)(7,15)(11,13)(12,16)(17,21,22,26)(18,20,23,30)(19,28,29,32)(24,27,25,31)(33,38,39,44)(34,42,43,48)(35,45,46,36)(37,47,40,41)

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=h,d^2=gh,e^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9113830);