#8416756

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	#55429	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3841 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)(17,18,21,22)(19,25,27,23)(20,24,28,29)(26,31,32,30)(33,34)(35,42)(36,37)(38,40)(39,43)(41,48)(44,45)(46,47);(1,2)(3,10)(4,12)(5,6)(7,9)(8,14)(11,15)(13,16)(18,22)(23,25)(24,29)(30,31)(33,35,39,38)(34,40,43,42)(36,46,41,44)(37,45,48,47);(2,6)(7,10)(8,12)(15,16)(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,29)(31,32)(33,36,39,41)(34,37,43,48)(35,44,38,46)(40,47,42,45);(1,3,5,9)(2,7,6,10)(4,13,14,11)(8,16,12,15)(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,29)(31,32)(33,37)(34,41)(35,45)(36,43)(38,47)(39,48)(40,44)(42,46);(1,4,5,14)(2,8,6,12)(3,11,9,13)(7,15,10,16)(17,19,21,27)(18,23,22,25)(20,26,28,32)(24,30,29,31)(33,35,39,38)(34,42,43,40)(36,44,41,46)(37,45,48,47);(1,2)(3,7)(4,8)(5,6)(9,10)(11,15)(12,14)(13,16)(17,20,21,28)(18,24,22,29)(19,26,27,32)(23,30,25,31)(33,38,39,35)(34,40,43,42)(36,46,41,44)(37,47,48,45)

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=gh,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, 8323952);