#7877204

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	#2669	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4350 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=gh,d^2=hi,e^2=gh,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,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, 7784400);