#7714543

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	#37130	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4212 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=h,d^2=g,e^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, 7621739);