#7898308

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	C₂×#548	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4280 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7805504);