#7879845

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	#42679	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4382 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=h,d^2=gh,e^2=h,f^2=i,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, 7787041);