#8497878

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	#37334	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4458 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)(9,10,14,18)(11,15,21,16)(12,17,23,13)(19,20,22,24)(25,26,30,31)(27,32,29,34)(28,36,37,33)(35,40,38,39);(3,7)(5,8)(9,11,14,21)(10,15,18,16)(12,22,23,19)(13,20,17,24)(26,31)(32,34)(33,36)(39,40);(1,2)(3,5)(4,6)(7,8)(9,11)(10,16)(12,19)(13,20)(14,21)(15,18)(17,24)(22,23)(25,27,30,29)(26,32,31,34)(28,35,37,38)(33,39,36,40);(1,3)(2,5)(4,7)(6,8)(9,11)(10,15)(12,19)(13,24)(14,21)(16,18)(17,20)(22,23)(25,28)(26,33)(27,35)(29,38)(30,37)(31,36)(32,39)(34,40);(1,2)(3,5)(4,6)(7,8)(9,12)(10,17)(11,19)(13,18)(14,23)(15,20)(16,24)(21,22)(25,26)(27,32)(28,33)(29,34)(30,31)(35,39)(36,37)(38,40);(9,13,14,17)(10,12,18,23)(11,20,21,24)(15,22,16,19)(25,29,30,27)(26,34,31,32)(28,35,37,38)(33,39,36,40)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8405074);