#7974967

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	#45968	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4474 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)(9,10,15,17)(11,16,14,18)(12,13,21,24)(19,20,22,23)(25,26)(27,32)(28,37)(29,40)(30,33)(31,34)(35,36)(38,39);(1,2,5,6)(3,7,4,8)(9,11,15,14)(10,16,17,18)(12,19,21,22)(13,20,24,23)(25,27,31,38)(26,32,34,39)(28,33,40,35)(29,36,37,30);(10,17)(12,21)(16,18)(19,22)(25,28,31,40)(26,29,34,37)(27,35,38,33)(30,39,36,32);(1,3,5,4)(2,7,6,8)(9,12)(10,13)(11,19)(14,22)(15,21)(16,20)(17,24)(18,23)(25,28)(26,29)(27,33)(30,39)(31,40)(32,36)(34,37)(35,38);(1,3,5,4)(2,7,6,8)(9,13)(10,12)(11,20)(14,23)(15,24)(16,19)(17,21)(18,22)(25,29)(26,28)(27,36)(30,38)(31,37)(32,33)(34,40)(35,39);(1,4,5,3)(2,8,6,7)(9,14,15,11)(10,18,17,16)(12,22,21,19)(13,23,24,20)(25,30,31,36)(26,33,34,35)(27,37,38,29)(28,39,40,32)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=i,d^2=g,e^2=g,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, 7882163);