#8372192

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

4442 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=g,d^2=gh,e^2=gh,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,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, 8279388);