#9267010

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	#40553	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3914 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=gi,d^2=i,e^2=h,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, 9174206);