#7720161

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	#36436	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4331 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,d^2=h,e^2=g,f^2=hi,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, 7627357);