#10324073

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

4548 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)(11,14)(12,15)(13,16)(17,19)(18,21)(20,22)(23,24)(25,26,30,34)(27,35,38,31)(28,39,37,32)(29,33,36,40);(1,2)(3,7)(4,5)(6,8)(10,11)(13,21)(15,17)(20,24)(25,27)(26,31)(28,36)(29,37)(30,38)(32,40)(33,39)(34,35);(2,5)(6,7)(12,19)(15,17)(20,24)(22,23)(25,28)(26,32)(27,29)(30,37)(31,33)(34,39)(35,40)(36,38);(2,5)(6,7)(9,11,14,10)(12,15,19,17)(13,18,21,16)(20,22,24,23)(25,28)(26,32)(27,36)(29,38)(30,37)(31,40)(33,35)(34,39);(1,2)(3,6)(4,5)(7,8)(9,12)(10,15)(11,17)(13,20)(14,19)(16,22)(18,23)(21,24)(25,29,30,36)(26,33,34,40)(27,37,38,28)(31,39,35,32);(1,3)(2,6)(4,8)(5,7)(9,13,14,21)(10,16,11,18)(12,20,19,24)(15,22,17,23)(25,26)(27,31)(28,32)(29,33)(30,34)(35,38)(36,40)(37,39)

Polycyclic presentation:

2^9|a^2=h,d^2=gh,e^2=h,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, 10231269);