#8739411

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	#42155	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4149 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,c^2=gh,d^2=gh,e^2=h,f^2=h,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, 8646607);