#8649011

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	C₂×#127	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4011 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=hi,d^2=ghi,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=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, 8556207);