#7940918

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	#2388	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4227 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)(11,14)(12,17)(13,16)(15,20)(18,22)(19,24)(21,23)(25,26,32,35)(27,36,31,30)(28,33,40,37)(29,38,39,34);(1,2)(3,5)(4,6)(7,8)(9,10,14,11)(12,15,20,17)(13,16,22,18)(19,23,21,24)(25,27,32,31)(26,30,35,36)(28,38,40,34)(29,37,39,33);(10,11)(15,17)(16,18)(23,24)(25,28)(26,33)(27,34)(29,36)(30,39)(31,38)(32,40)(35,37);(1,3,4,7)(2,5,6,8)(9,11,14,10)(12,17,20,15)(13,18,22,16)(19,24,21,23)(25,29,32,39)(26,34,35,38)(27,37,31,33)(28,36,40,30);(9,12)(10,15)(11,17)(13,21)(14,20)(16,24)(18,23)(19,22)(25,30,32,36)(26,27,35,31)(28,39,40,29)(33,34,37,38);(9,13,14,22)(10,16,11,18)(12,19,20,21)(15,23,17,24)(25,31,32,27)(26,30,35,36)(28,38,40,34)(29,33,39,37)

Polycyclic presentation:

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

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7848114);