#8691785

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

4211 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8598981);