#8092024

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, prime radical, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4013 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=gi,d^2=h,e^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=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, 7999220);