#8800213

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	#48505	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4441 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=g,d^2=i,e^2=gi,f^2=g,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, 8707409);