#8057916

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	#46080	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4177 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)(9,10,14,15)(11,16,19,12)(13,21,22,17)(18,20,24,23)(25,26)(27,34)(28,33)(29,30)(31,35)(32,38)(36,40)(37,39);(1,2)(3,7)(4,8)(5,6)(9,11,14,19)(10,12,15,16)(13,18,22,24)(17,20,21,23)(25,27)(26,32)(28,39)(29,37)(30,36)(31,38)(33,40)(34,35);(1,3,5,4)(2,7,6,8)(10,15)(12,16)(17,21)(20,23)(25,28,31,30)(26,33,35,29)(27,36,38,39)(32,37,34,40);(9,12,14,16)(10,11,15,19)(13,20,22,23)(17,18,21,24)(25,29)(26,28)(27,37)(30,35)(31,33)(32,39)(34,36)(38,40);(9,11)(10,16)(12,15)(13,18)(14,19)(17,23)(20,21)(22,24)(25,30,31,28)(26,29,35,33)(27,36,38,39)(32,37,34,40);(1,4,5,3)(2,8,6,7)(9,13)(10,17)(11,18)(12,20)(14,22)(15,21)(16,23)(19,24)(25,27)(26,34)(28,36)(29,37)(30,39)(31,38)(32,35)(33,40)

Polycyclic presentation:

2^9|a^2=h,b^2=h,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, 7965112);