#8773008

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	#44453	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4343 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)(3,4)(5,6)(7,8)(9,10,14,18)(11,12,16,15)(13,17,22,24)(19,20,21,23)(25,26,30,33)(27,29,32,38)(28,31,37,40)(34,36,35,39);(2,6)(4,7)(9,11,14,16)(10,15,18,12)(13,21,22,19)(17,20,24,23)(27,32)(29,38)(34,35)(36,39);(2,6)(3,8)(9,10,14,18)(11,15,16,12)(13,17,22,24)(19,23,21,20)(25,27,30,32)(26,29,33,38)(28,35,37,34)(31,39,40,36);(1,3,5,8)(2,7,6,4)(9,10)(11,15)(12,16)(13,17)(14,18)(19,23)(20,21)(22,24)(25,28)(26,31)(27,34)(29,36)(30,37)(32,35)(33,40)(38,39);(1,2)(3,7)(4,8)(5,6)(9,12,14,15)(10,16,18,11)(13,20,22,23)(17,21,24,19)(25,28)(26,31)(27,34)(29,36)(30,37)(32,35)(33,40)(38,39);(1,4,5,7)(2,8,6,3)(9,13)(10,17)(11,19)(12,20)(14,22)(15,23)(16,21)(18,24)(25,29)(26,32)(27,33)(28,36)(30,38)(31,35)(34,40)(37,39)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=ghi,d^2=h,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,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, 8680204);