#10320927

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

4660 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:

(9,10,14,15)(11,12,18,19)(13,16,21,22)(17,20,23,24)(25,26)(27,32)(28,30)(29,36)(31,35)(33,37)(34,38)(39,40);(1,2)(3,7)(4,5)(6,8)(10,15)(12,19)(13,21)(17,23)(25,26,30,28)(27,32,35,31)(29,33,37,36)(34,39,40,38);(2,5)(6,7)(9,11,14,18)(10,12,15,19)(13,17,21,23)(16,20,22,24)(25,27)(26,31)(28,32)(29,34)(30,35)(33,38)(36,39)(37,40);(2,5)(6,7)(9,12,14,19)(10,11,15,18)(13,20,21,24)(16,17,22,23)(25,28,30,26)(27,32,35,31)(29,36,37,33)(34,39,40,38);(1,2)(3,6)(4,5)(7,8)(9,11)(10,12)(13,17)(14,18)(15,19)(16,20)(21,23)(22,24)(25,27)(26,32)(28,31)(29,34)(30,35)(33,39)(36,38)(37,40);(1,3)(2,6)(4,8)(5,7)(9,13,14,21)(10,16,15,22)(11,17,18,23)(12,20,19,24)(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=h,c^2=gh,d^2=g,f^2=g,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, 10228123);