#8418375

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

4756 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=gh,d^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8325571);