#7972725

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	#2407	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4268 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)(11,19)(12,18)(13,16)(14,20)(15,17)(21,22)(23,24)(25,26,30,34)(27,35,36,31)(28,32,39,38)(29,33,40,37);(1,3,5,8)(2,6,4,7)(9,11,15,13)(10,16,17,19)(12,21,20,23)(14,22,18,24)(25,27,30,36)(26,31,34,35)(28,37,39,33)(29,32,40,38);(1,2,5,4)(3,6,8,7)(10,17)(12,20)(16,19)(21,23)(27,36)(28,39)(31,35)(32,38);(9,12,15,20)(10,18,17,14)(11,21,13,23)(16,24,19,22)(25,26)(27,31)(28,32)(29,33)(30,34)(35,36)(37,40)(38,39);(9,13,15,11)(10,19,17,16)(12,23,20,21)(14,24,18,22)(25,28)(26,32)(27,37)(29,35)(30,39)(31,40)(33,36)(34,38);(1,4,5,2)(3,7,8,6)(9,14,15,18)(10,20,17,12)(11,22,13,24)(16,23,19,21)(25,29,30,40)(26,33,34,37)(27,38,36,32)(28,31,39,35)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=ghi,d^2=h,e^2=h,f^2=g,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, 7879921);