#8417214

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) = 128

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=gh,d^2=i,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, 8324410);