#7674365

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

10726 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7581561);