#7934486

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	#2408	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4383 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)(11,18)(12,17)(13,14)(15,19)(16,22)(20,24)(21,23)(26,31)(27,33)(28,36)(38,40);(1,2,5,6)(3,7,4,8)(9,11)(10,16)(12,23)(13,21)(14,20)(15,22)(17,24)(18,19)(25,26)(27,34)(28,32)(29,33)(30,31)(35,40)(36,39)(37,38);(1,3,5,4)(2,7,6,8)(9,12,15,14)(10,17,19,13)(11,20,22,23)(16,21,18,24)(25,27,30,33)(26,29,31,34)(28,37,36,35)(32,40,39,38);(1,3,5,4)(2,7,6,8)(9,13)(10,12)(11,21)(14,19)(15,17)(16,23)(18,20)(22,24)(25,28,30,36)(26,32,31,39)(27,35,33,37)(29,38,34,40);(1,4,5,3)(2,8,6,7)(9,14,15,12)(10,13,19,17)(11,20,22,23)(16,21,18,24)(25,26)(27,29)(28,32)(30,31)(33,34)(35,38)(36,39)(37,40);(9,11)(10,18)(12,20)(13,21)(14,23)(15,22)(16,19)(17,24)(25,29)(26,33)(27,31)(28,38)(30,34)(32,37)(35,39)(36,40)

Polycyclic presentation:

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

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7841682);