#10322703

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

4313 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,d^2=g,e^2=g,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, 10229899);