#7806344

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4380 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7713540);