#8437813

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	#57584	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4388 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)(3,4)(5,6)(7,8)(10,15)(11,17)(12,19)(20,24)(25,26)(27,36)(28,32)(29,33)(30,34)(31,35)(37,38)(39,40);(2,6)(3,7)(9,10,14,15)(11,13,17,21)(12,16,19,22)(18,20,23,24)(27,28)(32,36)(37,39)(38,40);(1,3,5,7)(2,4,6,8)(25,27,31,28)(26,32,35,36)(29,37,34,39)(30,38,33,40);(1,4)(2,7)(3,6)(5,8)(9,11)(10,13)(12,18)(14,17)(15,21)(16,20)(19,23)(22,24)(25,28,31,27)(26,32,35,36)(29,39,34,37)(30,38,33,40);(9,12)(10,16)(11,18)(13,20)(14,19)(15,22)(17,23)(21,24)(25,29,31,34)(26,33,35,30)(27,37,28,39)(32,40,36,38);(9,13)(10,17)(11,15)(12,20)(14,21)(16,23)(18,22)(19,24)(25,30,31,33)(26,34,35,29)(27,38,28,40)(32,39,36,37)

Polycyclic presentation:

2^9|b^2=ghi,c^2=i,d^2=h,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,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, 8345009);