#7852755

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	#40723	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4555 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)(11,20)(12,17)(13,23)(14,18)(15,19)(16,22)(21,24)(25,26,32,30)(27,36,39,33)(28,29,34,38)(31,35,40,37);(1,2,5,6)(3,7,4,8)(9,11)(10,16)(12,14)(13,18)(15,22)(17,24)(19,20)(21,23)(25,27,32,39)(26,33,30,36)(28,35,34,37)(29,31,38,40);(9,12,15,23)(10,13,19,17)(11,21,22,14)(16,24,20,18)(25,28)(26,29)(27,37)(30,38)(31,36)(32,34)(33,40)(35,39);(1,3,5,4)(2,7,6,8)(9,12)(10,17)(11,14)(13,19)(15,23)(16,24)(18,20)(21,22)(25,29,32,38)(26,34,30,28)(27,31,39,40)(33,37,36,35);(1,4,5,3)(2,8,6,7)(9,13,15,17)(10,12,19,23)(11,18,22,24)(14,20,21,16)(25,30,32,26)(27,36,39,33)(28,38,34,29)(31,35,40,37);(9,14)(10,18)(11,12)(13,16)(15,21)(17,20)(19,24)(22,23)(25,31)(26,35)(27,38)(28,36)(29,39)(30,37)(32,40)(33,34)

Polycyclic presentation:

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

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7759951);