#462315

order	512
nilpotency class	3
isoclinism	isoclinic class of #455561
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2622
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#537	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

880 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 369511);