#7903857

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

4609 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)(11,13)(12,15)(14,17)(16,18)(19,21)(20,23)(22,24)(25,26)(27,35)(28,39)(29,34)(30,32)(31,38)(33,37)(36,40);(1,2)(3,6)(4,5)(7,8)(9,11)(10,13)(12,18)(14,19)(15,16)(17,21)(20,22)(23,24)(25,27)(26,31)(28,36)(29,37)(30,38)(32,35)(33,40)(34,39);(1,3,4,7)(2,6,5,8)(10,12)(11,16)(17,20)(19,24)(26,32)(31,35)(33,39)(34,40);(9,12,15,10)(11,18,16,13)(14,20,23,17)(19,22,24,21)(25,26)(27,31)(28,33)(29,34)(30,32)(35,38)(36,40)(37,39);(9,13,15,18)(10,16,12,11)(14,22,23,21)(17,19,20,24)(25,28,30,37)(26,33,32,39)(27,36,38,29)(31,40,35,34);(9,14,15,23)(10,17,12,20)(11,19,16,24)(13,21,18,22)(25,29,30,36)(26,34,32,40)(27,37,38,28)(31,39,35,33)

Polycyclic presentation:

2^9|c^2=ghi,d^2=i,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7811053);