#7866447

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	#45236	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4354 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)(9,10,14,12)(11,17,19,15)(13,16,21,20)(18,23,24,22)(25,26)(27,32)(28,29)(30,35)(31,33)(34,40)(36,37)(38,39);(1,2,5,6)(3,7,4,8)(9,11)(10,15)(12,17)(13,18)(14,19)(16,22)(20,23)(21,24)(25,27)(26,32)(28,36)(29,37)(30,33)(31,35)(34,39)(38,40);(1,3,5,4)(2,7,6,8)(10,12)(15,17)(16,20)(22,23)(26,33)(27,35)(28,34)(37,38);(9,12,14,10)(11,17,19,15)(13,20,21,16)(18,23,24,22)(25,28,31,34)(26,29,33,40)(27,36,35,39)(30,38,32,37);(9,11)(10,15)(12,17)(13,18)(14,19)(16,22)(20,23)(21,24)(25,29)(26,34)(27,37)(28,33)(30,36)(31,40)(32,39)(35,38);(1,4,5,3)(2,8,6,7)(9,13,14,21)(10,16,12,20)(11,18,19,24)(15,22,17,23)(25,30)(26,35)(27,33)(28,38)(29,39)(31,32)(34,37)(36,40)

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=ghi,d^2=h,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,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, 7773643);