#7865070

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

4254 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,d^2=h,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, 7772266);