#10372284

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

4514 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10279480);