#7865078

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

4188 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,d^2=gh,e^2=g,f^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7772274);