#9240948

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

3772 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=g,d^2=h,e^2=gh,f^2=h,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, 9148144);