#8722231

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

4039 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:

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

Polycyclic presentation:

2^9|a^2=ghi,c^2=h,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,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, 8629427);