#8295592

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	#40696	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4476 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)(9,10,14,15)(11,19,20,16)(12,22,23,17)(13,18,24,21);(2,6)(3,7)(10,15)(11,20)(17,22)(18,21)(25,26,30,31)(27,29,33,38)(28,32,37,39)(34,36,35,40);(1,3)(2,4)(5,7)(6,8)(10,15)(13,24)(16,19)(17,22)(26,31)(29,38)(32,39)(36,40);(1,4)(2,3)(5,8)(6,7)(9,11,14,20)(10,16,15,19)(12,18,23,21)(13,17,24,22)(25,27,30,33)(26,29,31,38)(28,35,37,34)(32,40,39,36);(9,12,14,23)(10,17,15,22)(11,21,20,18)(13,19,24,16)(25,28)(26,32)(27,34)(29,36)(30,37)(31,39)(33,35)(38,40);(9,13)(10,18)(11,17)(12,19)(14,24)(15,21)(16,23)(20,22)(25,29)(26,33)(27,31)(28,36)(30,38)(32,35)(34,39)(37,40)

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,d^2=gi,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=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, 8202788);