#6709206

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667772
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#42586	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1393 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,c^2=f,d^2=fgi,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^c=ei,h^c=hi,e^d=ei,h^e=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6616402);