#7735636

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	#50749	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4356 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)(11,16)(12,17)(13,18)(14,19)(15,20)(21,24)(22,23)(25,26,31,33)(27,32,29,37)(28,34,38,39)(30,35,40,36);(1,2)(3,7)(4,8)(5,6)(9,11)(10,15)(12,21)(13,17)(14,20)(16,19)(18,23)(22,24)(25,27,31,29)(26,32,33,37)(28,36,38,35)(30,39,40,34);(11,20)(12,23)(15,16)(17,22)(26,33)(27,29)(34,39)(35,36);(1,3,5,4)(2,7,6,8)(9,11)(10,16)(12,21)(13,22)(14,20)(15,19)(17,24)(18,23)(25,28,31,38)(26,34,33,39)(27,36,29,35)(30,37,40,32);(1,3,5,4)(2,7,6,8)(9,12,14,23)(10,17,19,22)(11,21,20,18)(13,16,24,15)(25,29,31,27)(26,32,33,37)(28,35,38,36)(30,39,40,34);(1,4,5,3)(2,8,6,7)(9,13,14,24)(10,18,19,21)(11,22,20,17)(12,16,23,15)(25,30)(26,35)(27,34)(28,37)(29,39)(31,40)(32,38)(33,36)

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^2=gi,e^2=gh,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,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, 7642832);