#8188559

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	#38608	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4413 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=gh,d^2=ghi,e^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8095755);