#7919653

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	#35361	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4617 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=hi,d^2=gh,e^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, 7826849);