#7932319

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	#2714	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4190 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)(9,10,16,20)(11,18,13,15)(12,19,14,17)(21,23,22,24)(25,26)(27,35)(28,29)(30,32)(31,34)(33,40)(36,37)(38,39);(1,2)(3,5)(4,6)(7,8)(9,11,16,13)(10,15,20,18)(12,22,14,21)(17,24,19,23);(9,12,16,14)(10,17,20,19)(11,21,13,22)(15,23,18,24)(25,27,31,30)(26,32,34,35)(28,38,33,36)(29,37,40,39);(9,13,16,11)(10,18,20,15)(12,21,14,22)(17,23,19,24)(25,28,31,33)(26,29,34,40)(27,36,30,38)(32,39,35,37);(9,14,16,12)(10,19,20,17)(11,21,13,22)(15,23,18,24)(25,29)(26,33)(27,37)(28,34)(30,39)(31,40)(32,36)(35,38);(1,3,4,7)(2,5,6,8)(9,15,16,18)(10,11,20,13)(12,23,14,24)(17,21,19,22)(25,30,31,27)(26,32,34,35)(28,38,33,36)(29,39,40,37)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=h,d^2=h,e^2=i,f^2=ghi,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, 7839515);