#8639904

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	#2456	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4336 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=gi,d^2=gh,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8547100);