#8899198

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

4319 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=g,d^2=gh,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8806394);