#8743420

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	#35979	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4211 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)(3,7,8,5)(9,10,14,18)(11,17,22,15)(12,13,23,16)(19,24,20,21)(25,26,31,30)(27,34,29,32)(28,33,36,38)(35,40,37,39);(1,3,4,8)(2,5,6,7)(9,11)(10,15)(12,19)(13,21)(14,22)(16,24)(17,18)(20,23)(28,36)(33,38)(35,37)(39,40);(1,3)(2,5)(4,8)(6,7)(9,12,14,23)(10,16,18,13)(11,19,22,20)(15,24,17,21)(25,27,31,29)(26,32,30,34)(28,35,36,37)(33,39,38,40);(1,3)(2,5)(4,8)(6,7)(9,12)(10,13)(11,20)(14,23)(15,24)(16,18)(17,21)(19,22)(25,28,31,36)(26,33,30,38)(27,35,29,37)(32,39,34,40);(1,2)(3,5)(4,6)(7,8)(9,13)(10,12)(11,21)(14,16)(15,19)(17,20)(18,23)(22,24)(25,29,31,27)(26,34,30,32)(28,37,36,35)(33,40,38,39);(9,11)(10,17)(12,19)(13,24)(14,22)(15,18)(16,21)(20,23)(25,30,31,26)(27,34,29,32)(28,38,36,33)(35,40,37,39)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=hi,d^2=gh,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8650616);