#8088164

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	#37788	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4326 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,d^2=gi,b^a=bg,c^a=ch,e^a=ei,c^b=ci,d^b=di,e^c=ei,f^c=fh,e^d=ehi,f^d=fi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7995360);