#8057932

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	#2714	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4039 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)(9,10,16,20)(11,17,22,23)(12,13,14,18)(15,21,24,19)(25,26)(27,34)(28,33)(29,38)(30,35)(31,36)(32,39)(37,40);(1,2)(3,7)(4,8)(5,6)(9,11,16,22)(10,17,20,23)(12,19,14,21)(13,15,18,24)(25,27)(26,32)(28,40)(29,37)(30,38)(31,39)(33,35)(34,36);(1,3,5,4)(2,7,6,8)(9,12,16,14)(10,18,20,13)(11,21,22,19)(15,23,24,17)(25,28,31,38)(26,33,36,29)(27,30,39,40)(32,37,34,35);(9,13,16,18)(10,12,20,14)(11,15,22,24)(17,19,23,21)(25,29)(26,28)(27,37)(30,34)(31,33)(32,40)(35,39)(36,38);(1,3,5,4)(2,7,6,8)(9,14,16,12)(10,18,20,13)(11,19,22,21)(15,23,24,17)(25,27)(26,34)(28,30)(29,37)(31,39)(32,36)(33,35)(38,40);(1,4,5,3)(2,8,6,7)(9,15)(10,19)(11,18)(12,23)(13,22)(14,17)(16,24)(20,21)(25,30)(26,35)(27,38)(28,39)(29,34)(31,40)(32,33)(36,37)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=gh,d^2=h,e^2=gi,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, 7965128);