#8807392

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	#39383	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4436 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=g,d^2=h,e^2=gh,f^2=gi,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, 8714588);