#8914524

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	#2670	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4311 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8821720);