#8393664

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	#44136	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4063 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)(4,12,14,8)(10,15,13,16)(17,18,23,21)(19,26,24,20)(22,25,30,29)(27,32,31,28);(1,3,6,11)(2,7,5,9)(4,10,14,13)(8,15,12,16)(17,19,23,24)(18,20,21,26)(22,27,30,31)(25,28,29,32)(33,34,38,39)(35,41,44,40)(36,45,47,37)(42,48,46,43);(2,5)(7,9)(8,12)(15,16)(22,30)(25,29)(27,31)(28,32)(34,39)(37,45)(40,41)(43,48);(1,4)(2,8)(3,10)(5,12)(6,14)(7,15)(9,16)(11,13)(17,20,23,26)(18,24,21,19)(22,28,30,32)(25,31,29,27)(33,35,38,44)(34,40,39,41)(36,46,47,42)(37,43,45,48);(1,3)(2,7)(4,13)(5,9)(6,11)(8,16)(10,14)(12,15)(17,21,23,18)(19,20,24,26)(22,29,30,25)(27,28,31,32)(33,36)(34,37)(35,42)(38,47)(39,45)(40,48)(41,43)(44,46);(1,5,6,2)(3,9,11,7)(4,12,14,8)(10,16,13,15)(17,22,23,30)(18,25,21,29)(19,27,24,31)(20,28,26,32)(33,37)(34,36)(35,43)(38,45)(39,47)(40,46)(41,42)(44,48)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,d^2=ghi,e^2=gh,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, 8300860);