#8806882

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3843 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=hi,d^2=hi,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8714078);