#7895146

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

4432 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=h,d^2=ghi,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7802342);