#7977472

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	#42107	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4359 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=ghi,e^2=g,f^2=h,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, 7884668);