#8679878

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

4165 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)(9,10,14,13)(11,17,19,15)(12,20,21,16)(18,22,24,23)(25,26)(27,33)(28,38)(29,35)(30,31)(32,36)(34,37)(39,40);(1,2,5,4)(3,7,8,6)(9,11)(10,15)(12,18)(13,17)(14,19)(16,22)(20,23)(21,24)(25,27,32,30)(26,33,36,31)(28,39,37,29)(34,35,38,40);(1,3,5,8)(2,6,4,7)(25,28)(26,34)(27,29)(30,39)(31,35)(32,37)(33,40)(36,38);(1,4,5,2)(3,7,8,6)(9,12,14,21)(10,16,13,20)(11,18,19,24)(15,22,17,23)(25,29,32,39)(26,35,36,40)(27,37,30,28)(31,38,33,34);(9,11)(10,17)(12,18)(13,15)(14,19)(16,23)(20,22)(21,24)(25,30,32,27)(26,33,36,31)(28,39,37,29)(34,40,38,35);(9,13,14,10)(11,17,19,15)(12,16,21,20)(18,22,24,23)(25,31,32,33)(26,30,36,27)(28,35,37,40)(29,34,39,38)

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,c^2=hi,d^2=gh,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi,f^e=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, 8587074);