#8389281

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	#36857	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4172 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=h,e^2=g,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, 8296477);