#7983167

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	#38938	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4444 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)(11,17)(15,21)(18,23)(22,24)(25,26)(27,33)(28,31)(29,34)(30,32)(35,40)(36,38)(37,39);(1,2)(3,7)(4,8)(5,6)(9,10)(11,15)(12,19)(13,14)(16,20)(17,21)(18,24)(22,23)(25,27,30,28)(26,31,32,33)(29,35,37,36)(34,38,39,40);(1,3,5,4)(2,7,6,8)(10,14)(15,21)(16,19)(22,24)(26,32)(31,33)(34,39)(38,40);(1,3,5,4)(2,7,6,8)(9,11)(10,15)(12,18)(13,17)(14,21)(16,22)(19,24)(20,23)(25,26)(27,31)(28,33)(29,34)(30,32)(35,38)(36,40)(37,39);(1,4,5,3)(2,8,6,7)(9,10)(11,15)(12,19)(13,14)(16,20)(17,21)(18,24)(22,23)(25,28,30,27)(26,33,32,31)(29,36,37,35)(34,40,39,38);(9,12,13,20)(10,16,14,19)(11,18,17,23)(15,22,21,24)(25,29,30,37)(26,34,32,39)(27,35,28,36)(31,38,33,40)

Polycyclic presentation:

2^9|b^2=h,c^2=gh,d^2=gh,e^2=g,f^2=hi,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, 7890363);