#7852769

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

4502 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)(9,10,15,14)(11,18,13,16)(12,17,22,21)(19,24,20,23)(25,26)(27,34)(28,33)(29,31)(30,32)(35,40)(36,38)(37,39);(1,2)(3,7)(4,8)(5,6)(9,11,15,13)(10,16,14,18)(12,19,22,20)(17,23,21,24)(25,27,31,30)(26,32,29,34)(28,35,38,37)(33,39,36,40);(9,11,15,13)(10,16,14,18)(12,20,22,19)(17,24,21,23)(25,26,31,29)(27,34,30,32)(28,36,38,33)(35,39,37,40);(9,12,15,22)(10,17,14,21)(11,19,13,20)(16,23,18,24)(25,28,31,38)(26,33,29,36)(27,35,30,37)(32,39,34,40);(1,3,5,4)(2,7,6,8)(9,13,15,11)(10,18,14,16)(12,20,22,19)(17,24,21,23)(25,29,31,26)(27,34,30,32)(28,36,38,33)(35,40,37,39);(1,4,5,3)(2,8,6,7)(9,14,15,10)(11,18,13,16)(12,21,22,17)(19,24,20,23)(25,30,31,27)(26,32,29,34)(28,37,38,35)(33,39,36,40)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=h,d^2=h,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, 7759965);