#8114412

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	#58553	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3847 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:

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

Polycyclic presentation:

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

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, 8021608);