#404311

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	#62856	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2374 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:

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

Polycyclic presentation:

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

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, 311507);