#8757360

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	#49585	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4198 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8664556);