#7711392

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	#37190	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4442 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)(11,15)(12,22)(13,23)(14,18)(16,20)(17,24)(19,21)(26,32)(30,40)(33,36)(34,38);(1,2)(3,6)(4,5)(7,8)(9,11,14,21)(10,15,18,19)(12,13,20,24)(16,17,22,23)(25,26)(27,36)(28,33)(29,34)(30,35)(31,32)(37,40)(38,39);(9,10,14,18)(11,19,21,15)(12,22,20,16)(13,17,24,23)(25,27,31,28)(26,33,32,36)(29,35,39,37)(30,34,40,38);(1,3,4,7)(2,6,5,8)(9,10)(11,15)(12,16)(13,17)(14,18)(19,21)(20,22)(23,24)(25,28,31,27)(26,33,32,36)(29,35,39,37)(30,38,40,34);(9,12,14,20)(10,16,18,22)(11,13,21,24)(15,17,19,23)(25,29,31,39)(26,34,32,38)(27,37,28,35)(30,36,40,33);(9,13)(10,17)(11,20)(12,21)(14,24)(15,22)(16,19)(18,23)(25,30)(26,35)(27,34)(28,38)(29,36)(31,40)(32,37)(33,39)

Polycyclic presentation:

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

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, 7618588);