#9685591

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	#39428	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4465 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=hi,d^2=g,e^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 9592787);