#8963154

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	#54955	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4056 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=ghi,c^2=g,d^2=gi,e^2=hi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,f^c=fgi,e^d=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, 8870350);