#7927689

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×#550	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4451 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)(3,7,8,5)(9,10)(11,22)(12,19)(13,24)(14,17)(15,20)(16,21)(18,23)(27,29)(32,34)(35,36)(39,40);(1,3,4,8)(2,5,6,7)(9,11)(10,15)(12,13)(14,23)(16,20)(17,24)(18,19)(21,22)(25,26,31,30)(27,32,29,34)(28,33,38,37)(35,39,36,40);(9,12,16,14)(10,17,21,19)(11,23,20,13)(15,18,22,24)(25,27,31,29)(26,32,30,34)(28,36,38,35)(33,40,37,39);(9,13,16,23)(10,18,21,24)(11,12,20,14)(15,19,22,17)(25,28,31,38)(26,33,30,37)(27,35,29,36)(32,39,34,40);(9,14,16,12)(10,19,21,17)(11,23,20,13)(15,18,22,24)(25,29,31,27)(26,34,30,32)(28,36,38,35)(33,40,37,39);(9,15,16,22)(10,20,21,11)(12,18,14,24)(13,19,23,17)(25,30,31,26)(27,34,29,32)(28,37,38,33)(35,40,36,39)

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=h,d^2=h,e^2=h,f^2=h,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, 7834885);