#8417040

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=gh,d^2=gi,e^2=i,f^2=g,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, 8324236);