#8081013

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	#47719	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4450 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)(11,19,12,16)(13,18,24,14)(20,21,22,23)(25,26,30,28)(27,33,34,31)(29,32,37,36)(35,39,40,38);(1,2)(3,7)(4,8)(5,6)(9,11,15,12)(10,16,17,19)(13,20,24,22)(14,21,18,23)(25,27)(26,31)(28,33)(29,35)(30,34)(32,38)(36,39)(37,40);(10,17)(14,18)(16,19)(21,23)(27,34)(31,33)(35,40)(38,39);(1,3,5,4)(2,7,6,8)(9,12,15,11)(10,16,17,19)(13,20,24,22)(14,23,18,21)(25,28,30,26)(27,31,34,33)(29,32,37,36)(35,39,40,38);(9,13)(10,18)(11,20)(12,22)(14,17)(15,24)(16,23)(19,21)(25,27)(26,31)(28,33)(29,35)(30,34)(32,38)(36,39)(37,40);(1,4,5,3)(2,8,6,7)(9,14,15,18)(10,13,17,24)(11,21,12,23)(16,20,19,22)(25,29,30,37)(26,32,28,36)(27,35,34,40)(31,38,33,39)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,d^2=g,f^2=g,b^a=bg,c^a=ch,e^a=ei,c^b=ci,d^b=di,e^c=ei,f^c=fh,e^d=ehi,f^d=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, 7988209);