#10424677

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	#41204	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4294 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=ghi,d^2=g,f^2=h,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, 10331873);