#9245131

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

3998 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,c^2=g,d^2=gh,e^2=h,f^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, 9152327);