#10511112

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

4555 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=gi,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10418308);