#8922827

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

4223 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:

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

Polycyclic presentation:

2^9|a^2=hi,d^2=ghi,e^2=i,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8830023);