#8720584

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
prime radical	#2679	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4050 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)(3,7)(4,12)(5,9)(6,10)(8,13)(11,14)(15,16)(18,22)(19,25)(24,29)(26,31)(33,34,39,40)(35,43,37,41)(36,46,47,42)(38,44,48,45);(3,10)(6,7)(11,16)(14,15)(17,18,21,22)(19,23,25,28)(20,24,27,29)(26,30,31,32)(34,40)(36,47)(41,43)(44,45);(1,3,5,10)(2,6,9,7)(4,11,13,16)(8,14,12,15)(17,18)(19,23)(20,24)(21,22)(25,28)(26,30)(27,29)(31,32)(33,35,39,37)(34,41,40,43)(36,44,47,45)(38,42,48,46);(1,3)(2,7)(4,11)(5,10)(6,9)(8,15)(12,14)(13,16)(17,18)(19,23)(20,24)(21,22)(25,28)(26,30)(27,29)(31,32)(33,36,39,47)(34,42,40,46)(35,45,37,44)(38,43,48,41);(1,4,5,13)(2,8,9,12)(3,11,10,16)(6,14,7,15)(17,19,21,25)(18,23,22,28)(20,26,27,31)(24,30,29,32)(33,37,39,35)(34,43,40,41)(36,44,47,45)(38,42,48,46);(1,2)(3,6)(4,12)(5,9)(7,10)(8,13)(11,15)(14,16)(17,20,21,27)(18,24,22,29)(19,26,25,31)(23,30,28,32)(33,38,39,48)(34,44,40,45)(35,42,37,46)(36,41,47,43)

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=gi,d^2=hi,e^2=h,f^2=g,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, 8627780);