#8195169

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	#2428	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4344 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,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=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, 8102365);