#8322140

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	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3806 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8229336);