#404309

order	512
nilpotency class	3
isoclinism	isoclinic class of #393982
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2846
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2319 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=ef,d^2=gi,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,e^b=ei,f^b=fi,h^b=hi,e^c=ei,h^c=hi,e^d=ei,f^d=fi

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, 311505);