#9843539

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	#56769	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4222 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=g,e^2=i,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, 9750735);