#7713882

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	#37454	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4446 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)(12,20)(16,21)(18,24)(22,23)(26,27)(29,37)(31,33)(35,40);(1,2)(3,6)(4,5)(7,8)(9,10)(11,17)(12,16)(13,14)(15,19)(18,23)(20,21)(22,24)(25,26,30,27)(28,31,36,33)(29,32,37,34)(35,38,40,39);(1,3,4,7)(2,6,5,8)(10,14)(15,17)(16,21)(22,23)(25,27,30,26)(28,31,36,33)(29,34,37,32)(35,38,40,39);(9,11,13,19)(10,15,14,17)(12,18,20,24)(16,22,21,23)(25,28)(26,31)(27,33)(29,35)(30,36)(32,38)(34,39)(37,40);(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,21)(23,24)(25,29)(26,32)(27,34)(28,35)(30,37)(31,38)(33,39)(36,40);(9,12)(10,16)(11,18)(13,20)(14,21)(15,22)(17,23)(19,24)

Polycyclic presentation:

2^9|b^2=h,c^2=g,d^2=i,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, 7621078);