#9240965

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3768 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)(3,4,5,9)(6,14,15,10)(11,16,13,12)(17,18,22,26)(19,23,27,32)(20,24,30,29)(21,25,31,28)(33,34)(35,43)(36,37)(38,40)(39,42)(41,48)(44,45)(46,47);(2,8)(3,5)(10,14)(11,13)(19,27)(21,31)(23,32)(25,28)(33,35,39,38)(34,40,42,43)(36,46,41,44)(37,45,48,47);(1,3,7,5)(2,9,8,4)(6,11,15,13)(10,16,14,12)(17,19)(18,23)(20,25)(21,29)(22,27)(24,31)(26,32)(28,30)(33,36,39,41)(34,37,42,48)(35,44,38,46)(40,47,43,45);(1,4,7,9)(2,3,8,5)(6,12,15,16)(10,11,14,13)(33,37)(34,41)(35,45)(36,42)(38,47)(39,48)(40,44)(43,46);(1,5,7,3)(2,9,8,4)(6,13,15,11)(10,16,14,12)(17,20)(18,24)(19,28)(21,23)(22,30)(25,27)(26,29)(31,32);(1,6)(2,10)(3,11)(4,12)(5,13)(7,15)(8,14)(9,16)(17,21)(18,25)(19,29)(20,23)(22,31)(24,27)(26,28)(30,32)(33,38,39,35)(34,40,42,43)(36,46,41,44)(37,47,48,45)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=g,d^2=h,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, 9148161);