#8417284

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

3977 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)(17,18,21,24)(19,22,25,28)(20,23,27,30)(26,29,31,32)(33,34,39,43)(35,41,45,38)(36,40,37,42)(44,46,47,48);(1,2,6,5)(3,7,11,9)(4,12,14,8)(10,16,13,15)(19,25)(22,28)(26,31)(29,32)(33,35,39,45)(34,38,43,41)(36,44,37,47)(40,48,42,46);(2,5)(7,9)(8,12)(15,16)(17,18)(19,22)(20,23)(21,24)(25,28)(26,29)(27,30)(31,32)(33,36,39,37)(34,40,43,42)(35,44,45,47)(38,48,41,46);(1,3)(2,7)(4,13)(5,9)(6,11)(8,16)(10,14)(12,15)(17,19,21,25)(18,22,24,28)(20,26,27,31)(23,29,30,32)(33,35)(34,41)(36,44)(37,47)(38,43)(39,45)(40,46)(42,48);(1,4,6,14)(2,8,5,12)(3,10,11,13)(7,15,9,16)(17,18)(19,22)(20,23)(21,24)(25,28)(26,29)(27,30)(31,32)(33,37,39,36)(34,42,43,40)(35,44,45,47)(38,48,41,46);(1,5,6,2)(3,9,11,7)(4,12,14,8)(10,16,13,15)(17,20,21,27)(18,23,24,30)(19,26,25,31)(22,29,28,32)(33,38,39,41)(34,35,43,45)(36,46,37,48)(40,47,42,44)

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=g,d^2=hi,e^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8324480);