#10324048

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
prime radical	#41057	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4269 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=g,d^2=gh,e^2=h,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, 10231244);