#10324052

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

4485 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)(11,17,19,13)(12,15,20,23)(18,24,21,22)(25,26)(27,32)(28,30)(29,36)(31,35)(33,37)(34,38)(39,40);(1,2)(3,7)(4,5)(6,8)(9,11,14,19)(10,13,16,17)(12,18,20,21)(15,22,23,24)(25,26,30,28)(27,32,35,31)(29,33,37,36)(34,39,40,38);(2,5)(6,7)(12,20)(15,23)(18,21)(22,24)(25,27)(26,31)(28,32)(29,34)(30,35)(33,38)(36,39)(37,40);(2,5)(6,7)(9,11)(10,13)(12,21)(14,19)(15,24)(16,17)(18,20)(22,23)(25,28,30,26)(27,32,35,31)(29,36,37,33)(34,39,40,38);(1,2)(3,6)(4,5)(7,8)(9,12,14,20)(10,15,16,23)(11,18,19,21)(13,22,17,24)(25,27)(26,32)(28,31)(29,34)(30,35)(33,39)(36,38)(37,40);(1,3)(2,6)(4,8)(5,7)(9,13)(10,11)(12,22)(14,17)(15,18)(16,19)(20,24)(21,23)(25,29,30,37)(26,33,28,36)(27,34,35,40)(31,38,32,39)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,d^2=h,e^2=gh,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, 10231248);