#8107579

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	#2428	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4053 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=gi,d^2=gi,e^2=h,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, 8014775);