#10412408

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	#37832	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4875 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=hi,d^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi

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, 10319604);