#9267037

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	#52531	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3838 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=gi,d^2=gi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dh,e^c=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, 9174233);