#7732426

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

4519 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,d^2=gh,e^2=hi,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=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, 7639622);