#8393618

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3978 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=h,d^2=i,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, 8300814);