#7716344

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	#44237	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4274 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7623540);