#10323054

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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=g,d^2=gh,e^2=g,f^2=h,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, 10230250);