#7955174

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	#2535	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4131 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)(9,10,15,14)(11,12,19,16)(13,21,22,17)(18,23,24,20)(25,26)(27,31)(28,35)(29,37)(30,33)(32,38)(34,36)(39,40);(1,3,5,8)(2,6,4,7)(9,11,15,19)(10,16,14,12)(13,18,22,24)(17,23,21,20)(25,27,30,35)(26,31,33,28)(29,34,38,40)(32,39,37,36);(1,2,5,4)(3,6,8,7)(9,12,15,16)(10,11,14,19)(13,20,22,23)(17,18,21,24)(25,28)(26,27)(29,36)(30,31)(32,34)(33,35)(37,40)(38,39);(1,4,5,2)(3,7,8,6)(9,12)(10,11)(13,20)(14,19)(15,16)(17,18)(21,24)(22,23)(25,26)(27,31)(28,35)(29,32)(30,33)(34,39)(36,40)(37,38);(9,13)(10,17)(11,18)(12,20)(14,21)(15,22)(16,23)(19,24)(25,29,30,38)(26,32,33,37)(27,34,35,40)(28,36,31,39);(9,14,15,10)(11,12,19,16)(13,17,22,21)(18,23,24,20)

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=g,d^2=gi,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, 7862370);