#7728226

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

4368 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)(9,10,15,14)(11,18,13,16)(12,17,22,21)(19,24,20,23)(25,26)(27,33)(28,31)(29,32)(30,35)(34,39)(36,38)(37,40);(1,3,5,8)(2,6,4,7)(9,11,15,13)(10,16,14,18)(12,19,22,20)(17,23,21,24)(25,27)(26,30)(28,34)(29,35)(31,37)(32,33)(36,40)(38,39);(11,13)(16,18)(19,20)(23,24)(25,26,29,32)(27,30,35,33)(28,31,36,38)(34,37,40,39);(9,12,15,22)(10,17,14,21)(11,19,13,20)(16,23,18,24)(25,26)(27,30)(28,31)(29,32)(33,35)(34,37)(36,38)(39,40);(9,13,15,11)(10,16,14,18)(12,20,22,19)(17,23,21,24)(25,27)(26,30)(28,34)(29,35)(31,37)(32,33)(36,40)(38,39);(1,4,5,2)(3,7,8,6)(9,14,15,10)(11,16,13,18)(12,21,22,17)(19,23,20,24)(25,28,29,36)(26,31,32,38)(27,34,35,40)(30,37,33,39)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=h,d^2=i,e^2=i,f^2=g,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, 7635422);