#7989552

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	C₂×#127	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4243 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=gi,d^2=h,e^2=gi,f^2=ghi,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, 7896748);