#7720348

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	#36374	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4394 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)(9,10)(11,16)(12,14)(13,20)(15,18)(17,21)(19,22)(23,24)(25,26,30,33)(27,28,31,35)(29,37,38,32)(34,40,39,36);(1,2,4,5)(3,6,7,8)(9,11)(10,15)(12,16)(13,19)(14,18)(17,22)(20,23)(21,24)(27,31)(28,35)(34,39)(36,40);(10,12)(11,18)(17,20)(19,24)(25,27)(26,28)(29,34)(30,31)(32,36)(33,35)(37,40)(38,39);(9,11)(10,16)(12,15)(13,19)(14,18)(17,23)(20,22)(21,24);(1,3,4,7)(2,6,5,8)(9,12,14,10)(11,15,18,16)(13,20,21,17)(19,22,24,23)(25,28)(26,31)(27,33)(29,36)(30,35)(32,39)(34,37)(38,40);(9,13,14,21)(10,17,12,20)(11,19,18,24)(15,22,16,23)(25,29,30,38)(26,32,33,37)(27,34,31,39)(28,36,35,40)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7627544);