#7902075

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

4120 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:

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

Polycyclic presentation:

2^9|c^2=g,d^2=h,e^2=ghi,f^2=i,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, 7809271);