#9862944

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	#40359	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4293 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,d^2=i,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9770140);