#8271092

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	#52164	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4435 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=gi,d^2=ghi,e^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi

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, 8178288);