#9039124

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	#49622	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3928 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8946320);