#8934890

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	#2818	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4045 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)(3,7)(4,13)(5,10)(6,11)(8,12)(9,15)(14,16)(17,18)(19,25)(20,23)(21,29)(22,24)(26,30)(27,31)(28,32)(33,34)(35,41)(36,43)(37,40)(38,42)(39,46)(44,48)(45,47);(1,3,6,12)(2,7,11,8)(4,14,15,10)(5,13,16,9)(17,19,22,20)(18,23,24,25)(21,27,30,28)(26,31,29,32)(33,35,40,39)(34,41,37,46)(36,45,38,44)(42,48,43,47);(3,12)(5,16)(7,8)(10,14)(18,24)(21,30)(23,25)(27,28)(33,36,40,38)(34,42,37,43)(35,44,39,45)(41,47,46,48);(1,3)(2,8)(4,14)(5,13)(6,12)(7,11)(9,16)(10,15)(17,18)(19,25)(20,23)(21,26)(22,24)(27,32)(28,31)(29,30)(33,37,40,34)(35,41,39,46)(36,42,38,43)(44,48,45,47);(1,4,6,15)(2,9,11,13)(3,10,12,14)(5,7,16,8)(17,20,22,19)(18,25,24,23)(21,28,30,27)(26,32,29,31)(33,38,40,36)(34,43,37,42)(35,44,39,45)(41,47,46,48);(1,5)(2,10)(3,13)(4,7)(6,16)(8,15)(9,12)(11,14)(17,21,22,30)(18,26,24,29)(19,27,20,28)(23,31,25,32)(33,39,40,35)(34,41,37,46)(36,44,38,45)(42,48,43,47)

Polycyclic presentation:

2^9|b^2=gi,c^2=gh,d^2=gh,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,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, 8842086);