#7942172

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	#39841	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4260 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=i,d^2=i,e^2=g,f^2=hi,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, 7849368);