#8393357

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	#43089	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4097 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)(3,10)(4,8)(5,14)(6,12)(7,9)(11,15)(13,16)(17,18)(19,25)(20,23)(21,24)(22,27)(26,31)(28,30)(29,32)(33,34)(35,39)(36,41)(37,45)(38,40)(42,48)(43,44)(46,47);(1,3,7,4)(2,8,9,10)(5,13,15,12)(6,11,16,14)(17,18,21,24)(19,25,27,22)(20,23,28,30)(26,31,32,29)(33,35,38,41)(34,39,40,36)(37,46,48,43)(42,44,45,47);(2,9)(6,16)(8,10)(11,14)(20,28)(23,30)(26,32)(29,31)(34,40)(35,41)(42,45)(43,46);(1,4,7,3)(2,10,9,8)(5,12,15,13)(6,14,16,11)(17,19,21,27)(18,22,24,25)(20,26,28,32)(23,29,30,31)(33,36,38,39)(34,41,40,35)(37,47,48,44)(42,43,45,46);(1,5,7,15)(2,11,9,14)(3,12,4,13)(6,10,16,8)(17,18)(19,25)(20,23)(21,24)(22,27)(26,31)(28,30)(29,32)(33,37,38,48)(34,42,40,45)(35,43,41,46)(36,44,39,47);(1,6,7,16)(2,12,9,13)(3,11,4,14)(5,10,15,8)(17,20,21,28)(18,23,24,30)(19,26,27,32)(22,29,25,31)(33,34)(35,39)(36,41)(37,42)(38,40)(43,47)(44,46)(45,48)

Polycyclic presentation:

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

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, 8300553);