#7898306

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	#2416	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4220 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=i,d^2=h,e^2=gh,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, 7805502);