#7648399

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	C₂³×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4448 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=i,d^2=g,f^2=hi,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^d=ehi

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, 7555595);