#8126988

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	#55222	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4315 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=i,d^2=h,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^c=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8034184);