#9173123

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4029 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=gh,d^2=gh,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, 9080319);