#8041140

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
prime radical	#40520	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4216 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)(11,14)(12,15)(13,18)(16,22)(17,20)(19,24)(21,23)(25,26,29,32)(27,30,35,39)(28,31,36,40)(33,37,34,38);(1,2)(3,6)(4,5)(7,8)(9,10,14,11)(12,17,20,15)(13,16,22,18)(19,24,21,23)(25,27)(26,30)(28,34)(29,35)(31,38)(32,39)(33,36)(37,40);(10,11)(15,17)(16,18)(23,24)(25,28)(26,31)(27,33)(29,36)(30,37)(32,40)(34,35)(38,39);(1,3,4,7)(2,6,5,8)(9,11,14,10)(12,17,20,15)(13,18,22,16)(19,24,21,23)(25,26)(27,30)(28,31)(29,32)(33,37)(34,38)(35,39)(36,40);(9,12,14,20)(10,15,11,17)(13,21,22,19)(16,24,18,23)(25,28)(26,31)(27,34)(29,36)(30,38)(32,40)(33,35)(37,39);(9,13)(10,16)(11,18)(12,19)(14,22)(15,23)(17,24)(20,21)(25,27)(26,30)(28,33)(29,35)(31,37)(32,39)(34,36)(38,40)

Polycyclic presentation:

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

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, 7948336);