#8745871

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

4133 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)(4,7)(5,9)(6,10)(11,13)(12,15)(14,16)(17,18,23,20)(19,27,29,24)(21,30,28,25)(22,26,32,31)(34,38)(35,41)(40,43)(42,44);(1,3)(2,5)(4,11)(6,9)(7,14)(8,10)(12,16)(13,15)(17,19,23,29)(18,24,20,27)(21,22,28,32)(25,26,30,31)(34,38)(39,46)(40,43)(47,48);(4,12)(7,15)(11,16)(13,14)(18,20)(24,27)(25,30)(26,31)(33,34,37,38)(35,39,41,46)(36,40,45,43)(42,47,44,48);(1,4)(2,7)(3,11)(5,14)(6,12)(8,13)(9,16)(10,15)(17,20,23,18)(19,24,29,27)(21,30,28,25)(22,26,32,31)(33,34)(35,39)(36,43)(37,38)(40,45)(41,46)(42,48)(44,47);(1,3)(2,8)(4,11)(5,10)(6,9)(7,13)(12,16)(14,15)(17,21)(18,25)(19,22)(20,30)(23,28)(24,26)(27,31)(29,32)(33,35)(34,39)(36,44)(37,41)(38,46)(40,48)(42,45)(43,47);(1,5,6,8)(2,9,10,3)(4,13,12,14)(7,11,15,16)(17,22)(18,26)(19,28)(20,31)(21,29)(23,32)(24,30)(25,27)(33,36,37,45)(34,40,38,43)(35,42,41,44)(39,47,46,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=i,d^2=hi,f^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8653067);