#7766332

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

4322 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=h,d^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, 7673528);