#8081029

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	#37810	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4368 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,d^2=g,e^2=g,f^2=g,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, 7988225);