#7712874

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

4523 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^b=di,d^c=dh,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, 7620070);