#7826712

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

4396 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)(25,26,30,35)(27,32,37,31)(28,36,39,33)(29,34,40,38);(1,3,5,8)(2,6,4,7)(9,10)(11,16)(12,13)(14,15)(17,22)(18,24)(19,21)(20,23)(25,26,30,35)(27,32,37,31)(28,33,39,36)(29,38,40,34);(10,15)(13,22)(16,23)(19,24)(25,27,30,37)(26,31,35,32)(28,34,39,38)(29,33,40,36);(1,2,5,4)(3,6,8,7)(9,11)(10,16)(12,21)(13,19)(14,20)(15,23)(17,18)(22,24)(25,27)(26,32)(28,38)(29,36)(30,37)(31,35)(33,40)(34,39);(1,2,5,4)(3,6,8,7)(9,12,14,17)(10,13,15,22)(11,18,20,21)(16,24,23,19)(25,28,30,39)(26,33,35,36)(27,34,37,38)(29,31,40,32);(1,4,5,2)(3,7,8,6)(9,13)(10,17)(11,19)(12,15)(14,22)(16,18)(20,24)(21,23)(25,29)(26,34)(27,36)(28,31)(30,40)(32,39)(33,37)(35,38)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7733908);