#7941746

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	#36426	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4418 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=gh,e^2=hi,f^2=i,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, 7848942);