#7879846

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

4352 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)(9,10,16,22)(11,17,13,19)(12,20,14,18)(15,21,24,23)(25,26)(27,34)(28,32)(29,31)(30,33)(35,40)(36,38)(37,39);(1,3,5,8)(2,6,4,7)(9,11,16,13)(10,17,22,19)(12,23,14,21)(15,18,24,20)(25,27,30,29)(26,31,33,34)(28,35,37,36)(32,38,39,40);(9,12,16,14)(10,18,22,20)(11,21,13,23)(15,19,24,17)(25,26)(27,34)(28,32)(29,31)(30,33)(35,40)(36,38)(37,39);(1,4,5,2)(3,7,8,6)(9,13,16,11)(10,19,22,17)(12,21,14,23)(15,20,24,18)(25,28,30,37)(26,32,33,39)(27,35,29,36)(31,38,34,40);(9,14,16,12)(10,20,22,18)(11,21,13,23)(15,19,24,17)(25,26)(27,31)(28,32)(29,34)(30,33)(35,38)(36,40)(37,39);(9,15,16,24)(10,21,22,23)(11,18,13,20)(12,19,14,17)(25,29,30,27)(26,31,33,34)(28,36,37,35)(32,38,39,40)

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=h,d^2=gh,e^2=h,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7787042);