#7971687

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	#2555	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4286 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)(9,10,15,12)(11,16,14,18)(13,20,22,17)(19,24,21,23)(25,26,31,34)(27,28,29,32)(30,33,39,40)(35,36,37,38);(1,2,5,6)(3,7,4,8)(9,11,15,14)(10,16,12,18)(13,19,22,21)(17,23,20,24)(25,27,31,29)(26,32,34,28)(30,37,39,35)(33,36,40,38);(1,3,5,4)(2,7,6,8)(10,12)(16,18)(17,20)(23,24)(25,28,31,32)(26,27,34,29)(30,36,39,38)(33,35,40,37);(9,12,15,10)(11,18,14,16)(13,20,22,17)(19,24,21,23)(25,29,31,27)(26,28,34,32)(30,37,39,35)(33,36,40,38);(1,3,5,4)(2,7,6,8)(9,13,15,22)(10,17,12,20)(11,19,14,21)(16,23,18,24)(25,28)(26,27)(29,34)(30,38)(31,32)(33,37)(35,40)(36,39);(1,4,5,3)(2,8,6,7)(9,14,15,11)(10,18,12,16)(13,21,22,19)(17,24,20,23)(25,30)(26,33)(27,35)(28,36)(29,37)(31,39)(32,38)(34,40)

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=g,d^2=h,e^2=gh,f^2=gh,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, 7878883);