#8591996

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	#2410	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4402 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=h,d^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8499192);