#7979797

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4225 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=g,d^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, 7886993);