#9675007

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	#51974	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4199 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=h,d^2=gh,e^2=g,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, 9582203);