#7865065

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

4219 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=gh,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, 7772261);