#8734942

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	#59735	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4287 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=gh,d^2=g,e^2=gh,f^2=g,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, 8642138);