#8932740

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	#46398	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4510 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,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,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, 8839936);