#7743454

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
prime radical	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3986 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=i,d^2=i,e^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,e^c=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7650650);