#7745334

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×#538	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4435 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,d^2=g,e^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,e^b=ei,d^c=dh,f^c=fi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7652530);