#7714746

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	#2424	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4154 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=i,e^2=i,f^2=g,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^b=di,d^c=dh,e^c=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7621942);