#7852777

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

4457 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=h,d^2=gh,e^2=g,f^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, 7759973);