#8416753

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	#55313	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3845 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=h,d^2=gi,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=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, 8323949);