#9490110

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

3705 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:

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

Polycyclic presentation:

2^9|b^2=hi,c^2=gh,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,d^c=dh,e^c=egi,f^c=fgi,e^d=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, 9397306);