#7901368

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	#51410	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4335 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)(9,10,14,16)(11,13,18,15)(12,20,22,17)(19,21,24,23)(25,26)(27,36)(28,29)(30,35)(31,33)(32,38)(34,39)(37,40);(1,3,5,8)(2,6,4,7)(9,11,14,18)(10,15,16,13)(12,19,22,24)(17,21,20,23)(25,27,31,38)(26,32,33,36)(28,35,34,40)(29,37,39,30);(1,2,5,4)(3,6,8,7)(10,16)(11,18)(17,20)(19,24)(26,33)(28,34)(32,36)(35,40);(1,2,5,4)(3,6,8,7)(9,10)(11,15)(12,17)(13,18)(14,16)(19,21)(20,22)(23,24)(25,28,31,34)(26,29,33,39)(27,35,38,40)(30,32,37,36);(1,2,5,4)(3,6,8,7)(9,12)(10,17)(11,19)(13,23)(14,22)(15,21)(16,20)(18,24)(25,29)(26,34)(27,37)(28,33)(30,38)(31,39)(32,40)(35,36);(1,4,5,2)(3,7,8,6)(9,13,14,15)(10,18,16,11)(12,21,22,23)(17,19,20,24)(25,30,31,37)(26,35,33,40)(27,29,38,39)(28,32,34,36)

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=gh,d^2=gi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=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, 7808564);