#9726115

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	#54018	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3952 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)(3,8)(4,13)(5,10)(6,14)(7,9)(11,16)(12,15)(17,18)(19,25)(20,21)(22,24)(23,29)(26,31)(27,30)(28,32)(36,44)(39,46)(43,47)(45,48);(1,3,7,4)(2,8,9,13)(5,14,15,11)(6,12,16,10)(17,19,22,29)(18,23,24,25)(20,27,26,32)(21,28,31,30)(33,34)(35,42)(36,39)(37,40)(38,41)(43,48)(44,46)(45,47);(2,9)(3,4)(10,12)(11,14)(18,24)(20,26)(23,25)(27,32)(33,35,38,40)(34,37,41,42)(36,43,44,47)(39,45,46,48);(1,4,7,3)(2,8,9,13)(5,14,15,11)(6,10,16,12)(17,20,22,26)(18,21,24,31)(19,27,29,32)(23,28,25,30)(33,36)(34,39)(35,43)(37,45)(38,44)(40,47)(41,46)(42,48);(1,5,7,15)(2,10,9,12)(3,11,4,14)(6,8,16,13)(17,19)(18,25)(20,27)(21,30)(22,29)(23,24)(26,32)(28,31)(33,34)(35,37)(36,39)(38,41)(40,42)(43,45)(44,46)(47,48);(1,6,7,16)(2,11,9,14)(3,12,4,10)(5,13,15,8)(17,21)(18,26)(19,28)(20,24)(22,31)(23,32)(25,27)(29,30)(33,37)(34,40)(35,41)(36,45)(38,42)(39,47)(43,46)(44,48)

Polycyclic presentation:

2^9|b^2=h,c^2=ghi,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=dh,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, 9633311);