#9160829

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	#44112	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4003 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)(3,7)(4,12)(5,6)(8,14)(9,11)(10,16)(13,15)(17,18,22,26)(19,27,29,23)(20,28,30,24)(21,25,32,31)(33,34,40,37)(35,41,45,39)(36,38,42,47)(43,48,46,44);(1,2)(3,9)(4,8)(5,6)(7,11)(10,16)(12,14)(13,15)(17,19,22,29)(18,23,26,27)(20,25,30,31)(21,24,32,28)(33,35,40,45)(34,41,37,39)(36,46,42,43)(38,44,47,48);(2,6)(7,9)(8,12)(15,16)(17,20,22,30)(18,24,26,28)(19,25,29,31)(21,23,32,27)(33,36,40,42)(34,38,37,47)(35,43,45,46)(39,44,41,48);(1,2)(3,9)(4,8)(5,6)(7,11)(10,16)(12,14)(13,15)(17,20)(18,24)(19,25)(21,27)(22,30)(23,32)(26,28)(29,31)(33,37,40,34)(35,39,45,41)(36,38,42,47)(43,48,46,44);(1,3,5,11)(2,7,6,9)(4,13,14,10)(8,16,12,15)(17,21)(18,25)(19,28)(20,23)(22,32)(24,29)(26,31)(27,30)(33,38)(34,42)(35,44)(36,37)(39,46)(40,47)(41,43)(45,48);(1,4,5,14)(2,8,6,12)(3,10,11,13)(7,15,9,16)(17,18)(19,23)(20,24)(21,31)(22,26)(25,32)(27,29)(28,30)(33,39)(34,35)(36,44)(37,45)(38,43)(40,41)(42,48)(46,47)

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=hi,d^2=ghi,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,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, 9068025);