#8772599

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	#2403	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4472 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=gi,d^2=h,e^2=h,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, 8679795);