#8057854

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	#2407	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4152 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:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=gh,d^2=h,e^2=i,f^2=ghi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7965050);