#9243815

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3715 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=h,d^2=g,e^2=gh,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, 9151011);