#8091728

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

4371 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)(9,10)(11,15)(12,16)(13,14)(17,23)(18,21)(19,24)(20,22)(25,26)(27,28)(29,33)(30,31)(32,34)(35,36)(37,39)(38,40);(1,3,5,8)(2,6,4,7)(25,27)(26,28)(29,35)(30,34)(31,32)(33,36)(37,40)(38,39);(10,14)(15,21)(16,22)(23,24)(26,31)(27,34)(29,37)(36,38);(1,2,5,4)(3,6,8,7)(9,11,13,18)(10,15,14,21)(12,19,20,17)(16,24,22,23)(25,28,30,32)(26,27,31,34)(29,38,37,36)(33,40,39,35);(1,2,5,4)(3,6,8,7)(9,12,13,20)(10,16,14,22)(11,17,18,19)(15,23,21,24)(25,27)(26,32)(28,31)(29,35)(30,34)(33,38)(36,39)(37,40);(1,4,5,2)(3,7,8,6)(9,10)(11,15)(12,16)(13,14)(17,23)(18,21)(19,24)(20,22)(25,29,30,37)(26,33,31,39)(27,35,34,40)(28,36,32,38)

Polycyclic presentation:

2^9|a^2=g,b^2=g,d^2=gh,e^2=ghi,f^2=gi,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, 7998924);