#8445597

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

4354 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=g,e^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8352793);