#7973924

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	#51977	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4354 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)(11,20)(12,13)(14,19)(15,17)(16,22)(18,23)(21,24)(25,26,31,34)(27,28,29,32)(30,33,39,40)(35,36,37,38);(1,2)(3,7)(4,8)(5,6)(9,11,15,22)(10,16,17,20)(12,19,18,24)(13,21,23,14)(25,27,31,29)(26,32,34,28)(30,37,39,35)(33,36,40,38);(10,17)(12,18)(16,20)(19,24)(25,28,31,32)(26,27,34,29)(30,36,39,38)(33,35,40,37);(1,3,5,4)(2,7,6,8)(9,12,15,18)(10,13,17,23)(11,19,22,24)(14,16,21,20)(25,29,31,27)(26,28,34,32)(30,37,39,35)(33,36,40,38);(9,13)(10,18)(11,21)(12,17)(14,22)(15,23)(16,24)(19,20)(25,28)(26,27)(29,34)(30,38)(31,32)(33,37)(35,40)(36,39);(1,4,5,3)(2,8,6,7)(9,14,15,21)(10,19,17,24)(11,13,22,23)(12,16,18,20)(25,30,31,39)(26,33,34,40)(27,35,29,37)(28,36,32,38)

Polycyclic presentation:

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

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7881120);