#7732050

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	#35569	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4360 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)(27,36)(28,31)(29,34)(30,33)(32,35)(37,40)(38,39);(1,2,5,6)(3,7,4,8)(9,11,15,13)(10,16,17,23)(12,19,21,22)(14,24,18,20)(25,27,32,30)(26,33,35,36)(28,39,29,37)(31,40,34,38);(1,3,5,4)(2,7,6,8)(10,17)(11,13)(12,21)(20,24)(25,28,32,29)(26,31,35,34)(27,37,30,39)(33,38,36,40);(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)(26,34,35,31)(27,37,30,39)(33,38,36,40);(9,13,15,11)(10,16,17,23)(12,22,21,19)(14,24,18,20)(25,30,32,27)(26,33,35,36)(28,37,29,39)(31,40,34,38);(9,14,15,18)(10,12,17,21)(11,20,13,24)(16,22,23,19)(25,31,32,34)(26,28,35,29)(27,38,30,40)(33,37,36,39)

Polycyclic presentation:

2^9|a^2=h,b^2=gh,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, 7639246);