#10368429

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4033 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=hi,d^2=ghi,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi

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, 10275625);