#7986003

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	#47862	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4435 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7893199);