#8464182

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	#42923	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4388 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8371378);