#8821944

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	#48741	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3930 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)(19,28)(20,31)(21,22)(23,24)(25,29)(26,30)(27,32)(33,34,39,41)(35,40,36,44)(37,38,47,48)(42,46,45,43);(1,2)(3,6)(4,12)(5,8)(7,13)(9,11)(10,15)(14,16)(18,24)(19,29)(21,27)(26,31)(33,35,39,36)(34,40,41,44)(37,45,47,42)(38,43,48,46);(1,2,5,8)(3,9,11,6)(4,7,13,12)(10,15,16,14)(17,19)(18,25)(20,27)(21,30)(22,31)(23,29)(24,28)(26,32)(33,36,39,35)(34,40,41,44)(37,45,47,42)(38,46,48,43);(1,2)(3,6)(4,7)(5,8)(9,11)(10,14)(12,13)(15,16)(17,20,23,30)(18,26,24,31)(19,27,29,21)(22,25,32,28);(1,3,5,11)(2,6,8,9)(4,10,13,16)(7,14,12,15)(17,21,23,27)(18,22,24,32)(19,30,29,20)(25,31,28,26)(33,37)(34,38)(35,42)(36,45)(39,47)(40,46)(41,48)(43,44);(1,4,5,13)(2,7,8,12)(3,10,11,16)(6,14,9,15)(17,22)(18,27)(19,31)(20,25)(21,24)(23,32)(26,29)(28,30)(33,38)(34,37)(35,43)(36,46)(39,48)(40,45)(41,47)(42,44)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=hi,d^2=g,e^2=i,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8729140);