#7806827

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	#40584	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4836 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7714023);