#7732049

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	#37280	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4326 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=g,d^2=g,e^2=i,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,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, 7639245);