#7768045

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

4564 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=gh,d^2=h,e^2=g,f^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,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, 7675241);