#8188048

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	#34967	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4087 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)(3,8)(4,10)(5,9)(6,11)(7,12)(13,14)(15,16)(17,18,22,21)(19,25,27,23)(20,28,29,24)(26,30,32,31)(33,34,40,44)(35,45,47,41)(36,48,46,39)(37,43,38,42);(1,3,7,5)(2,8,12,9)(4,15,11,13)(6,14,10,16)(17,18,22,21)(19,25,27,23)(20,24,29,28)(26,31,32,30)(33,35)(34,41)(36,37)(38,46)(39,43)(40,47)(42,48)(44,45);(1,3,7,5)(2,9,12,8)(4,13,11,15)(6,14,10,16)(17,19,22,27)(18,23,21,25)(20,26,29,32)(24,30,28,31)(33,36,40,46)(34,39,44,48)(35,37,47,38)(41,43,45,42);(1,4,7,11)(2,10,12,6)(3,13,5,15)(8,14,9,16)(33,37,40,38)(34,42,44,43)(35,36,47,46)(39,41,48,45);(1,5,7,3)(2,8,12,9)(4,15,11,13)(6,16,10,14)(17,20,22,29)(18,24,21,28)(19,26,27,32)(23,30,25,31)(33,38,40,37)(34,43,44,42)(35,46,47,36)(39,45,48,41);(1,6,7,10)(2,11,12,4)(3,14,5,16)(8,13,9,15)(17,21,22,18)(19,23,27,25)(20,28,29,24)(26,30,32,31)(33,39)(34,36)(35,43)(37,41)(38,45)(40,48)(42,47)(44,46)

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=h,d^2=hi,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

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, 8095244);