#7971830

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

4366 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=gi,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,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, 7879026);