#7876788

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	#38280	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4430 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=g,d^2=hi,e^2=gh,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,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, 7783984);