#8932997

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	#46447	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4460 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,d^2=g,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8840193);