#8656195

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

4306 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,d^2=hi,e^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,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, 8563391);