#7972330

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	#2543	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4259 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)(11,19,21,16)(13,22,24,17)(14,18,20,23)(25,26,31,34)(27,28,29,32)(30,33,39,40)(35,36,37,38);(1,2,5,6)(3,7,4,8)(9,11,15,21)(10,16,12,19)(13,20,24,14)(17,23,22,18)(25,27,31,29)(26,32,34,28)(30,37,39,35)(33,36,40,38);(10,12)(16,19)(17,22)(18,23)(25,28,31,32)(26,27,34,29)(30,36,39,38)(33,35,40,37);(1,3,5,4)(2,7,6,8)(9,12,15,10)(11,19,21,16)(13,22,24,17)(14,23,20,18)(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)(10,17)(11,20)(12,22)(14,21)(15,24)(16,23)(18,19)(25,28)(26,27)(29,34)(30,38)(31,32)(33,37)(35,40)(36,39);(1,4,5,3)(2,8,6,7)(9,14,15,20)(10,18,12,23)(11,13,21,24)(16,17,19,22)(25,30,31,39)(26,33,34,40)(27,35,29,37)(28,36,32,38)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=i,d^2=g,e^2=gh,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, 7879526);