#9498385

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	#49881	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3849 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)(33,34,39,41)(35,42,37,36)(38,40,47,48)(43,46,45,44);(1,3)(2,8)(4,11)(5,7)(6,10)(9,15)(12,14)(13,16)(17,18,22,27)(19,25,28,23)(20,24,30,31)(21,26,32,29)(33,35,39,37)(34,36,41,42)(38,43,47,45)(40,44,48,46);(2,6)(3,5)(4,12)(15,16)(17,19,22,28)(18,23,27,25)(20,26,30,29)(21,31,32,24)(33,36,39,42)(34,35,41,37)(38,46,47,44)(40,45,48,43);(1,4,7,12)(2,9,6,13)(3,11,5,14)(8,15,10,16)(17,20)(18,24)(19,26)(21,23)(22,30)(25,32)(27,31)(28,29)(33,34)(35,36)(37,42)(38,40)(39,41)(43,44)(45,46)(47,48);(1,5,7,3)(2,10,6,8)(4,11,12,14)(9,15,13,16)(17,19)(18,25)(20,29)(21,24)(22,28)(23,27)(26,30)(31,32)(33,37,39,35)(34,36,41,42)(38,45,47,43)(40,44,48,46);(1,6,7,2)(3,10,5,8)(4,13,12,9)(11,16,14,15)(17,21)(18,26)(19,24)(20,23)(22,32)(25,30)(27,29)(28,31)(33,38,39,47)(34,40,41,48)(35,43,37,45)(36,44,42,46)

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=ghi,d^2=h,e^2=gh,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, 9405581);