#7896009

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4463 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,d^2=gi,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=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, 7803205);