#8107588

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	#2678	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4064 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8014784);