#7989528

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×#127	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4339 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7896724);