#9183909

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	#2456	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3882 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)(3,6)(4,8)(5,9)(7,11)(10,14)(12,13)(15,16)(17,18,22,26)(19,27,28,23)(20,24,30,32)(21,25,29,31)(33,34,39,36)(35,42,38,40)(37,41,46,44)(43,48,45,47);(1,3)(2,6)(4,10)(5,11)(7,9)(8,14)(12,15)(13,16)(17,18)(19,23)(20,24)(21,31)(22,26)(25,29)(27,28)(30,32)(33,34,39,36)(35,40,38,42)(37,44,46,41)(43,48,45,47);(1,3,5,11)(2,6,9,7)(4,10,13,16)(8,14,12,15)(17,19,22,28)(18,23,26,27)(20,29,30,21)(24,25,32,31)(33,35,39,38)(34,40,36,42)(37,43,46,45)(41,47,44,48);(1,3)(2,7)(4,10)(5,11)(6,9)(8,15)(12,14)(13,16)(17,20,22,30)(18,24,26,32)(19,21,28,29)(23,31,27,25)(33,36,39,34)(35,42,38,40)(37,44,46,41)(43,48,45,47);(1,2)(3,6)(4,12)(5,9)(7,11)(8,13)(10,15)(14,16)(17,21)(18,25)(19,20)(22,29)(23,32)(24,27)(26,31)(28,30)(33,37,39,46)(34,41,36,44)(35,43,38,45)(40,47,42,48);(1,4)(2,8)(3,10)(5,13)(6,14)(7,15)(9,12)(11,16)(17,18)(19,23)(20,24)(21,31)(22,26)(25,29)(27,28)(30,32)(33,38,39,35)(34,42,36,40)(37,45,46,43)(41,48,44,47)

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=hi,d^2=h,e^2=gh,f^2=gh,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, 9091105);