#9498349

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	#53228	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3842 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)(3,8)(4,9)(5,10)(6,7)(11,15)(12,13)(14,16)(17,18,22,26)(19,27,29,23)(20,24,28,30)(21,25,31,32);(1,3)(2,8)(4,11)(5,7)(6,10)(9,15)(12,14)(13,16)(17,19,22,29)(18,23,26,27)(20,21,28,31)(24,32,30,25)(33,34,38,43)(35,41,44,39)(36,40,46,47)(37,42,48,45);(2,6)(3,5)(4,12)(15,16)(20,28)(21,31)(24,30)(25,32)(33,35,38,44)(34,39,43,41)(36,42,46,45)(37,47,48,40);(1,4,7,12)(2,9,6,13)(3,11,5,14)(8,15,10,16)(17,18)(19,23)(20,30)(21,25)(22,26)(24,28)(27,29)(31,32)(33,36)(34,40)(35,42)(37,39)(38,46)(41,48)(43,47)(44,45);(1,5,7,3)(2,10,6,8)(4,11,12,14)(9,15,13,16)(17,20,22,28)(18,24,26,30)(19,21,29,31)(23,32,27,25)(33,35)(34,41)(36,45)(37,40)(38,44)(39,43)(42,46)(47,48);(1,6,7,2)(3,10,5,8)(4,13,12,9)(11,16,14,15)(17,21)(18,25)(19,28)(20,29)(22,31)(23,24)(26,32)(27,30)(33,37)(34,42)(35,40)(36,39)(38,48)(41,46)(43,45)(44,47)

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=hi,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,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, 9405545);