#8696061

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

4185 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=i,e^2=h,f^2=g,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, 8603257);